Recent zbMATH articles in MSC 60Ghttps://www.zbmath.org/atom/cc/60G2021-04-16T16:22:00+00:00WerkzeugCache miss estimation for non-stationary request processes.https://www.zbmath.org/1456.680202021-04-16T16:22:00+00:00"Olmos, Felipe"https://www.zbmath.org/authors/?q=ai:olmos.felipe"Graham, Carl"https://www.zbmath.org/authors/?q=ai:graham.carl"Simonian, Alain"https://www.zbmath.org/authors/?q=ai:simonian.alain-dSummary: The goal of the paper is to evaluate the miss probability of a Least Recently Used (LRU) cache, when it is offered a non-stationary request process given by a Poisson cluster point process. First, we construct a probability space using Palm theory, describing how to consider a tagged document with respect to the rest of the request process. This framework allows us to derive a fundamental integral formula for the expected number of misses of the tagged document. Then, we consider the limit when the cache size and the arrival rate go to infinity in proportion, and use the integral formula to derive an asymptotic expansion of the miss probability in powers of the inverse of the cache size. This enables us to quantify and improve the accuracy of the so-called \textit{Che approximation}.Large deviations for correlated random variables described by a matrix product ansatz.https://www.zbmath.org/1456.600622021-04-16T16:22:00+00:00"Angeletti, Florian"https://www.zbmath.org/authors/?q=ai:angeletti.florian"Touchette, Hugo"https://www.zbmath.org/authors/?q=ai:touchette.hugo"Bertin, Eric"https://www.zbmath.org/authors/?q=ai:bertin.eric-m"Abry, Patrice"https://www.zbmath.org/authors/?q=ai:abry.patriceLifschitz tail for alloy-type models driven by the fractional Laplacian.https://www.zbmath.org/1456.601192021-04-16T16:22:00+00:00"Kaleta, Kamil"https://www.zbmath.org/authors/?q=ai:kaleta.kamil"Pietruska-Pałuba, Katarzyna"https://www.zbmath.org/authors/?q=ai:pietruska-paluba.katarzynaSummary: We establish precise asymptotics near zero of the integrated density of states for the random Schrödinger operators \(( - \Delta )^{\alpha / 2} + V^\omega\) in \(L^2( \mathbb{R}^d)\) for the full range of \(\alpha \in(0, 2]\) and a fairly large class of random nonnegative alloy-type potentials \(V^\omega \). The IDS exhibits the Lifschitz tail singularity. We prove the existence of the limit \[\lim_{\lambda \to 0} \lambda^{d / \alpha} \ln N(\lambda) = - C \omega_d \left( \lambda_d^{( \alpha )} \right)^{d / \alpha},\] with \(C \in(0, \infty]\). The constant \(C\) is finite if and only if the common distribution of the lattice random variables charges \(\{0\}\). In this case, the constant \(C\) is expressed explicitly in terms of this distribution. In the limit formula, \( \lambda_d^{( \alpha )}\) denotes the Dirichlet ground-state eigenvalue of the operator \(( - \Delta )^{\alpha / 2}\) in the unit ball in \(\mathbb{R}^d\), and \(\omega_d\) is the volume of this ball.Nonlinear Brownian dynamics of interfacial fluctuations in a shear flow.https://www.zbmath.org/1456.760532021-04-16T16:22:00+00:00"Thiébaud, Marine"https://www.zbmath.org/authors/?q=ai:thiebaud.marine"Amarouchene, Yacine"https://www.zbmath.org/authors/?q=ai:amarouchene.yacine"Bickel, Thomas"https://www.zbmath.org/authors/?q=ai:bickel.thomasInfinitely ramified point measures and branching Lévy processes.https://www.zbmath.org/1456.602252021-04-16T16:22:00+00:00"Bertoin, Jean"https://www.zbmath.org/authors/?q=ai:bertoin.jean"Mallein, Bastien"https://www.zbmath.org/authors/?q=ai:mallein.bastienIn analogy to the well-known relation between infinitely divisible distributions and processes with stationary independent increments (Lévy processes), the authors connect what they call infinitely ramified point measures (IRPM) with branching Lévy processes (BLP). An IRPM is defined as a random point measure \(\mathcal{Z}\) which for every \(n\in\mathbb{N}\) has the same distribution as the \(n\)th generation of some branching random walk. In the considered BLP, particles move independently according to Lévy process and produce progeny during their lifetime similarly as in a Crump-Mode-Jagers branching process. The point measures, random walks and Lévy processes are taken here on the real line. Denote
\(\langle\Sigma_n \delta_{x_n},f\rangle:= \Sigma_n f(x_n)\) and \(\textbf{e}_\theta(x) :=\textbf{e}^{x\theta}\), \(\theta\ge 0\), \(x\in\mathbb{R}\). It is shown that given an IRPM \(\mathcal{Z}\) such that
(*) \(0< E(\langle\mathcal{Z}, \textbf{e}_\theta\rangle)<\infty\)
for some \(\theta\ge 0\), there exists a BLP \(Z= \{Z_t; t\ge 0\}\) with \(\mathcal{Z}\overset{(d)}{=} Z_1\). Vice versa, if \(Z\) is a BLP such that the corresponding Lévy measure satisfies certain integrability conditions, then \(Z_1\) is an IRPM satisfying (*).
Reviewer: Heinrich Hering (Rockenberg)Strong existence and higher order Fréchet differentiability of stochastic flows of fractional Brownian motion driven SDEs with singular drift.https://www.zbmath.org/1456.601402021-04-16T16:22:00+00:00"Baños, David"https://www.zbmath.org/authors/?q=ai:banos.david-r"Nilssen, Torstein"https://www.zbmath.org/authors/?q=ai:nilssen.torstein-k"Proske, Frank"https://www.zbmath.org/authors/?q=ai:proske.frank-norbertSummary: In this paper we present a new method for the construction of strong solutions of SDE's with merely integrable drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter \(H<\frac{1}{2}\). Furthermore, we prove the rather surprising result of the higher order Fréchet differentiability of stochastic flows of such SDE's in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a ``local time variational calculus''. We expect that our general approach can be also applied to the study of certain types of stochastic partial differential equations as e.g. stochastic conservation laws driven by rough paths.Rare events and scaling properties in field-induced anomalous dynamics.https://www.zbmath.org/1456.828302021-04-16T16:22:00+00:00"Burioni, R."https://www.zbmath.org/authors/?q=ai:burioni.raffaella"Gradenigo, G."https://www.zbmath.org/authors/?q=ai:gradenigo.giacomo"Sarracino, A."https://www.zbmath.org/authors/?q=ai:sarracino.alessandro"Vezzani, A."https://www.zbmath.org/authors/?q=ai:vezzani.alessandro"Vulpiani, A."https://www.zbmath.org/authors/?q=ai:vulpiani.angeloTime-uniform Chernoff bounds via nonnegative supermartingales.https://www.zbmath.org/1456.600542021-04-16T16:22:00+00:00"Howard, Steven R."https://www.zbmath.org/authors/?q=ai:howard.steven-r"Ramdas, Aaditya"https://www.zbmath.org/authors/?q=ai:ramdas.aaditya-k"McAuliffe, Jon"https://www.zbmath.org/authors/?q=ai:mcauliffe.jon-d"Sekhon, Jasjeet"https://www.zbmath.org/authors/?q=ai:sekhon.jasjeet-sSummary: We develop a class of exponential bounds for the probability that a martingale sequence crosses a time-dependent linear threshold. Our key insight is that it is both natural and fruitful to formulate exponential concentration inequalities in this way. We illustrate this point by presenting a single assumption and theorem that together unify and strengthen many tail bounds for martingales, including classical inequalities (1960--80) by Bernstein, Bennett, Hoeffding, and Freedman; contemporary inequalities (1980--2000) by Shorack and Wellner, Pinelis, Blackwell, van de Geer, and de la Peña; and several modern inequalities (post-2000) by Khan, Tropp, Bercu and Touati, Delyon, and others. In each of these cases, we give the strongest and most general statements to date, quantifying the time-uniform concentration of scalar, matrix, and Banach-space-valued martingales, under a variety of nonparametric assumptions in discrete and continuous time. In doing so, we bridge the gap between existing line-crossing inequalities, the sequential probability ratio test, the Cramér-Chernoff method, self-normalized processes, and other parts of the literature.Space-fractional Fokker-Planck equation and optimization of random search processes in the presence of an external bias.https://www.zbmath.org/1456.827842021-04-16T16:22:00+00:00"Palyulin, Vladimir V."https://www.zbmath.org/authors/?q=ai:palyulin.vladimir-v"Chechkin, Aleksei V."https://www.zbmath.org/authors/?q=ai:chechkin.aleksei-v"Metzler, Ralf"https://www.zbmath.org/authors/?q=ai:metzler.ralfCorrelation function for the grid-Poisson Euclidean matching on a line and on a circle.https://www.zbmath.org/1456.910642021-04-16T16:22:00+00:00"Boniolo, Elena"https://www.zbmath.org/authors/?q=ai:boniolo.elena"Caracciolo, Sergio"https://www.zbmath.org/authors/?q=ai:caracciolo.sergio"Sportiello, Andrea"https://www.zbmath.org/authors/?q=ai:sportiello.andreaOn the martingale decompositions of Gundy, Meyer, and Yoeurp in infinite dimensions.https://www.zbmath.org/1456.601002021-04-16T16:22:00+00:00"Yaroslavtsev, Ivan S."https://www.zbmath.org/authors/?q=ai:yaroslavtsev.ivanIt is proved that \(X\) is a UMD Banach space if and only if every local martingale \(M: {R}_+ \times \Omega \to X\) has a decomposition \(M=M^{c}+M^{q}+M^{a}\), where \(M^{c}\) is a continuous local martingale, \(M^{q}\) is a purely discontinuous quasi-left continuous local martingale and \(M^{a}\) is a purely discontinuous local martingale with accessible jumps. Moreover, the weak type inequalities
\[
\lambda P(N_t^{*}> \lambda) \leq C E(N_t)
\]
are shown in all the three cases \(N=M^{c},M^{q},M^{a}\).
Reviewer: Ferenc Weisz (Budapest)On intermediate level sets of two-dimensional discrete Gaussian free field.https://www.zbmath.org/1456.600822021-04-16T16:22:00+00:00"Biskup, Marek"https://www.zbmath.org/authors/?q=ai:biskup.marek"Louidor, Oren"https://www.zbmath.org/authors/?q=ai:louidor.orenSummary: We consider the discrete Gaussian free field (DGFF) in scaled-up (square-lattice) versions of suitably regular continuum domains \(D\subset\mathbb{C}\) and describe the scaling limit, including local structure, of the level sets at heights growing as a \(\lambda \)-multiple of the height of the absolute maximum, for any \(\lambda\in(0,1)\). We prove that, in the scaling limit, the scaled spatial position of a typical point \(x\) sampled from this level set is distributed according to a Liouville Quantum Gravity (LQG) measure in \(D\) at parameter equal \(\lambda \)-times its critical value, the field value at \(x\) has an exponential intensity measure and the configuration near \(x\) reduced by the value at \(x\) has the law of a pinned DGFF reduced by a suitable multiple of the potential kernel. In particular, the law of the total size of the level set, properly-normalized, converges to that of the total mass of the LQG measure. This sharpens considerably an earlier conclusion by
\textit{O. Daviaud} [Ann. Probab. 34, No. 3, 962--986 (2006; Zbl 1104.60062)].On moving-average models with feedback.https://www.zbmath.org/1456.622042021-04-16T16:22:00+00:00"Li, Dong"https://www.zbmath.org/authors/?q=ai:li.dong.2"Ling, Shiqing"https://www.zbmath.org/authors/?q=ai:ling.shiqing"Tong, Howell"https://www.zbmath.org/authors/?q=ai:tong.howellSummary: Moving average models, linear or nonlinear, are characterized by their short memory. This paper shows that, in the presence of feedback in the dynamics, the above characteristic can disappear.Integral fluctuation relations for entropy production at stopping times.https://www.zbmath.org/1456.600962021-04-16T16:22:00+00:00"Neri, Izaak"https://www.zbmath.org/authors/?q=ai:neri.izaak"Roldán, Édgar"https://www.zbmath.org/authors/?q=ai:roldan.edgar"Pigolotti, Simone"https://www.zbmath.org/authors/?q=ai:pigolotti.simone"Jülicher, Frank"https://www.zbmath.org/authors/?q=ai:julicher.frankStochastic resonance in a non-Poissonian dichotomous process: a new analytical approach.https://www.zbmath.org/1456.600912021-04-16T16:22:00+00:00"Bologna, Mauro"https://www.zbmath.org/authors/?q=ai:bologna.mauro"Chandía, Kristopher J."https://www.zbmath.org/authors/?q=ai:chandia.kristopher-j"Tellini, Bernardo"https://www.zbmath.org/authors/?q=ai:tellini.bernardoEstimation of the lead-lag parameter from non-synchronous data.https://www.zbmath.org/1456.622482021-04-16T16:22:00+00:00"Hoffmann, M."https://www.zbmath.org/authors/?q=ai:hoffmann.marc-r"Rosenbaum, M."https://www.zbmath.org/authors/?q=ai:rosenbaum.mathieu"Yoshida, N."https://www.zbmath.org/authors/?q=ai:yoshida.nozomu|yoshida.norio|yoshida.naoki|yoshida.naoshi|yoshida.norimasa|yoshida.noriyoshi|yoshida.naohiro|yoshida.nakahiro|yoshida.naofumi|yoshida.natsumi|yoshida.noriaki|yoshida.norinobu|yoshida.nobuyuki|yoshida.naoya|yoshida.naoto|yoshida.nobuakiSummary: We propose a simple continuous time model for modeling the lead-lag effect between two financial assets. A two-dimensional process \((X_{t},Y_{t})\) reproduces a lead-lag effect if, for some time shift \(\vartheta \in \mathbb{R} \), the process \((X_{t},Y_{t+\vartheta})\) is a semi-martingale with respect to a certain filtration. The value of the time shift \(\vartheta\) is the lead-lag parameter. Depending on the underlying filtration, the standard no-arbitrage case is obtained for \(\vartheta=0\). We study the problem of estimating the unknown parameter \(\vartheta \in \mathbb{R}\), given randomly sampled non-synchronous data from \((X_{t})\) and \((Y_{t})\). By applying a certain contrast optimization based on a modified version of the Hayashi-Yoshida covariation estimator, we obtain a consistent estimator of the lead-lag parameter, together with an explicit rate of convergence governed by the sparsity of the sampling design.The Jarzynski relation, fluctuation theorems, and stochastic thermodynamics for non-Markovian processes.https://www.zbmath.org/1456.825712021-04-16T16:22:00+00:00"Speck, T."https://www.zbmath.org/authors/?q=ai:speck.thomas"Seifert, U."https://www.zbmath.org/authors/?q=ai:seifert.udoRotational invariance of stochastic processes with application to fractional dynamics.https://www.zbmath.org/1456.600742021-04-16T16:22:00+00:00"Ślezak, Jakub K."https://www.zbmath.org/authors/?q=ai:slezak.jakub-karol"Magdziarz, Marcin"https://www.zbmath.org/authors/?q=ai:magdziarz.marcinA proof of Sznitman's conjecture about ballistic RWRE.https://www.zbmath.org/1456.602712021-04-16T16:22:00+00:00"Guerra, Enrique"https://www.zbmath.org/authors/?q=ai:guerra.enrique"Ramírez, Alejandro F."https://www.zbmath.org/authors/?q=ai:ramirez.alejandro-fSummary: We consider random walk in a uniformly elliptic i.i.d. random environment in \(\mathbb{Z}^d\) for \(d \geq 2\). It is believed that whenever the random walk is transient in a given direction it is necessarily ballistic. In order to quantify the gap which would be needed to prove this equivalence, several ballisticity conditions have been introduced. In particular, Sznitman defined the so-called conditions \((T)\) and \((T')\). The first one is the requirement that certain unlikely exit probabilities from a set of slabs decay exponentially fast with their width \(L\). The second one is the requirement that for all \(\gamma \in (0, 1)\) condition \((T)_\gamma\) is satisfied, which in turn is defined as the requirement that the decay is like \(e^{-CL^\gamma}\) for some \(C > 0\). In this article we prove a conjecture of \textit{A.-S. Sznitman} from [Probab. Theory Relat. Fields 122, No. 4, 509--544 (2002; Zbl 0995.60097)], stating that \((T)\) and \((T')\) are equivalent. Hence, this closes the circle proving the equivalence of conditions \((T), (T')\), and \((T)_\gamma\) for some \(\gamma \in (0, 1)\) as conjectured by Sznitman, and also of each of these ballisticity conditions with the polynomial condition \((P)_M\) for \(M \geq 15d + 5\) introduced by \textit{N. Berger} et al. in [Commun. Pure Appl. Math. 67, No. 12, 1947--1973 (2014; Zbl 1364.60140)].Asymmetric Lévy flights in nonhomogeneous environments.https://www.zbmath.org/1456.827952021-04-16T16:22:00+00:00"Srokowski, Tomasz"https://www.zbmath.org/authors/?q=ai:srokowski.tomaszA relativistically covariant random walk.https://www.zbmath.org/1456.828282021-04-16T16:22:00+00:00"Almaguer, J."https://www.zbmath.org/authors/?q=ai:almaguer.juan-antonio"Larralde, H."https://www.zbmath.org/authors/?q=ai:larralde.hernanModeling anomalous diffusion by a subordinated fractional Lévy-stable process.https://www.zbmath.org/1456.602782021-04-16T16:22:00+00:00"Teuerle, Marek"https://www.zbmath.org/authors/?q=ai:teuerle.marek-a"Wyłomańska Agnieszka"https://www.zbmath.org/authors/?q=ai:wylomanska-agnieszka."Sikora, Grzegorz"https://www.zbmath.org/authors/?q=ai:sikora.grzegorzMallows permutations and finite dependence.https://www.zbmath.org/1456.600812021-04-16T16:22:00+00:00"Holroyd, Alexander E."https://www.zbmath.org/authors/?q=ai:holroyd.alexander-e"Hutchcroft, Tom"https://www.zbmath.org/authors/?q=ai:hutchcroft.tom"Levy, Avi"https://www.zbmath.org/authors/?q=ai:levy.aviSummary: We use the Mallows permutation model to construct a new family of stationary finitely dependent proper colorings of the integers. We prove that these colorings can be expressed as finitary factors of i.i.d. processes with finite mean coding radii. They are the first colorings known to have these properties. Moreover, we prove that the coding radii have exponential tails, and that the colorings can also be expressed as functions of countable-state Markov chains. We deduce analogous existence statements concerning shifts of finite type and higher-dimensional colorings.Linear fractional stable motion: A wavelet estimator of the \(\alpha\) parameter.https://www.zbmath.org/1456.621772021-04-16T16:22:00+00:00"Ayache, Antoine"https://www.zbmath.org/authors/?q=ai:ayache.antoine"Hamonier, Julien"https://www.zbmath.org/authors/?q=ai:hamonier.julienSummary: Linear fractional stable motion, denoted by \(\{X_{H,\alpha }(t)\}_{t\in \mathbb R}\), is one of the most classical stable processes; it depends on two parameters H\(\in \)(0,1) and \(\alpha \in \)(0,2). The parameter H characterizes the self-similarity property of \(\{X_{H,\alpha }(t)\}_{t\in \mathbb R}\) while the parameter \(\alpha \) governs the tail heaviness of its finite dimensional distributions; throughout our article we assume that the latter distributions are symmetric, that \(H>1/\alpha\) and that H is known. We show that, on the interval [0,1], the asymptotic behavior of the maximum, at a given scale j, of absolute values of the wavelet coefficients of \(\{X_{H,\alpha }(t)\}_{t\in \mathbb R}\), is of the same order as \(2^{ - j(H - 1/\alpha )}\); then we derive from this result a strongly consistent (i.e. almost surely convergent) statistical estimator for the parameter \(\alpha \).Spatial and temporal white noises under sublinear \(G\)-expectation.https://www.zbmath.org/1456.601332021-04-16T16:22:00+00:00"Ji, Xiaojun"https://www.zbmath.org/authors/?q=ai:ji.xiaojun.1|ji.xiaojun"Peng, Shige"https://www.zbmath.org/authors/?q=ai:peng.shigeSummary: Under the framework of sublinear expectation, we introduce a new type of \(G\)-Gaussian random fields, which contains a type of spatial white noise as a special case. Based on this result, we also introduce a spatial-temporal \(G\)-white noise. Different from the case of linear expectation, in which the probability measure needs to be known, under the uncertainty of probability measures, spatial white noises are intrinsically different from temporal cases.Tightness for the minimal displacement of branching random walk.https://www.zbmath.org/1456.602262021-04-16T16:22:00+00:00"Bramson, Maury"https://www.zbmath.org/authors/?q=ai:bramson.maury-d"Zeitouni, Ofer"https://www.zbmath.org/authors/?q=ai:zeitouni.oferRecords in a changing world.https://www.zbmath.org/1456.600492021-04-16T16:22:00+00:00"Krug, Joachim"https://www.zbmath.org/authors/?q=ai:krug.joachimOn a semilinear double fractional heat equation driven by fractional Brownian sheet.https://www.zbmath.org/1456.601632021-04-16T16:22:00+00:00"Xia, Dengfeng"https://www.zbmath.org/authors/?q=ai:xia.dengfeng"Yan, Litan"https://www.zbmath.org/authors/?q=ai:yan.litan"Yin, Xiuwei"https://www.zbmath.org/authors/?q=ai:yin.xiuweiSummary: In this paper, we consider the stochastic heat equation of the form
\[ \frac{\partial u}{\partial t}=(\Delta_\alpha+\Delta_\beta)u+ \frac{\partial f}{\partial x}(t,x,u)+\frac{\partial^2W}{\partial t\partial x}, \]
where \(1<\beta<\alpha<2\), \(W(t,x)\) is a fractional Brownian sheet, \(\Delta_\theta:=-(-\Delta)^{\theta/2}\) denotes the fractional Lapalacian operator and \(f:[0,T]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) is a nonlinear measurable function. We introduce the existence, uniqueness and Hölder regularity of the solution. As a related question, we consider also a large deviation principle associated with the above equation with a small perturbation via an equivalence relationship between Laplace principle and large deviation principle.Mean unknotting times of random knots and embeddings.https://www.zbmath.org/1456.824322021-04-16T16:22:00+00:00"Chan, Yao-Ban"https://www.zbmath.org/authors/?q=ai:chan.yao-ban"Owczarek, Aleksander L."https://www.zbmath.org/authors/?q=ai:owczarek.aleksander-l"Rechnitzer, Andrew"https://www.zbmath.org/authors/?q=ai:rechnitzer.andrew-daniel"Slade, Gordon"https://www.zbmath.org/authors/?q=ai:slade.gordonStationary points in coalescing stochastic flows on \(\mathbb{R}\).https://www.zbmath.org/1456.601142021-04-16T16:22:00+00:00"Dorogovtsev, Andrey A."https://www.zbmath.org/authors/?q=ai:dorogovtsev.andrey-a"Riabov, Georgii V."https://www.zbmath.org/authors/?q=ai:riabov.georgii-v"Schmalfuß, Björn"https://www.zbmath.org/authors/?q=ai:schmalfuss.bjornSummary: This work is devoted to long-time properties of the Arratia flow with drift -- a stochastic flow on \(\mathbb{R}\) whose one-point motions are weak solutions to a stochastic differential equation \(dX(t) = a(X(t))dt + dw(t)\) that move independently before the meeting time and coalesce at the meeting time. We study special modification of such flow that gives rise to a random dynamical system and thus allows to discuss stationary points. Existence of a unique stationary point is proved in the case of a strictly monotone Lipschitz drift by developing a variant of a pullback procedure. Connections between the existence of a stationary point and properties of a dual flow are discussed.Regularity, continuity and approximation of isotropic Gaussian random fields on compact two-point homogeneous spaces.https://www.zbmath.org/1456.601222021-04-16T16:22:00+00:00"Cleanthous, Galatia"https://www.zbmath.org/authors/?q=ai:cleanthous.galatia"Georgiadis, Athanasios G."https://www.zbmath.org/authors/?q=ai:georgiadis.athanasios-g"Lang, Annika"https://www.zbmath.org/authors/?q=ai:lang.annika"Porcu, Emilio"https://www.zbmath.org/authors/?q=ai:porcu.emilioSummary: Gaussian random fields defined over compact two-point homogeneous spaces are considered and Sobolev regularity and Hölder continuity are explored through spectral representations. It is shown how spectral properties of the covariance function associated to a given Gaussian random field are crucial to determine such regularities and geometric properties. Furthermore, fast approximations of random fields on compact two-point homogeneous spaces are derived by truncation of the series expansion, and a suitable bound for the error involved in such an approximation is provided.High excursions of Bessel and related random processes.https://www.zbmath.org/1456.600882021-04-16T16:22:00+00:00"Piterbarg, Vladimir I."https://www.zbmath.org/authors/?q=ai:piterbarg.vladimir-i"Rodionov, Igor V."https://www.zbmath.org/authors/?q=ai:rodionov.i-vSummary: Asymptotic behavior of large excursions probabilities is evaluated for Euclidean norm of a wide class of Gaussian non-stationary vector processes with independent identically distributed components. It is assumed that the components have means zero and variances reaching its absolute maximum at only one point of the considered time interval. The Bessel process is an important example of such processes.The speed of a general random walk reinforced by its recent history.https://www.zbmath.org/1456.601092021-04-16T16:22:00+00:00"Pinsky, Ross G."https://www.zbmath.org/authors/?q=ai:pinsky.ross-gSummary: We consider a class of random walks whose increment distributions depend on the average value of the process over its most recent \(N\) steps. We investigate the speed of the process, and in particular, the limiting speed as the ``history window'' \(N \to \infty\).Malliavin smoothness on the Lévy space with Hölder continuous or \(B V\) functionals.https://www.zbmath.org/1456.601152021-04-16T16:22:00+00:00"Laukkarinen, Eija"https://www.zbmath.org/authors/?q=ai:laukkarinen.eijaSummary: We consider Malliavin smoothness of random variables \(f(X_1)\), where \(X\) is a pure jump Lévy process and the function \(f\) is either bounded and Hölder continuous or of bounded variation. We show that Malliavin differentiability and fractional differentiability of \(f(X_1)\) depend both on the regularity of \(f\) and the Blumenthal-Getoor index of the Lévy measure.Backward stochastic differential equations with two barriers and generalized reflection.https://www.zbmath.org/1456.601442021-04-16T16:22:00+00:00"Falkowski, Adrian"https://www.zbmath.org/authors/?q=ai:falkowski.adrian"Słomiński, Leszek"https://www.zbmath.org/authors/?q=ai:slominski.leszekSummary: We prove the existence and uniqueness of solutions of backward stochastic differential equations (BSDEs) with generalized reflection at time dependent càdlàg barriers. The reflection model we consider includes, as special cases, the standard reflection as well as the mirror reflection studied earlier in the theory of forward stochastic differential equations. We also show that the solution of BSDEs with generalized reflection corresponds to the value of an optimal stopping problem.Anomalous transport in disordered exclusion processes with coupled particles.https://www.zbmath.org/1456.602582021-04-16T16:22:00+00:00"Juhász, Róbert"https://www.zbmath.org/authors/?q=ai:juhasz.robertLarge deviation inequalities of LS estimator in nonlinear regression models.https://www.zbmath.org/1456.620462021-04-16T16:22:00+00:00"Miao, Yu"https://www.zbmath.org/authors/?q=ai:miao.yu"Tang, Yanyan"https://www.zbmath.org/authors/?q=ai:tang.yanyanSummary: In the paper, the large deviation inequalities of the LS estimator for the nonlinear regression model with martingale differences errors are established. The assumptions for the errors are (conditional) exponential integrability which weaken the bounded condition in [\textit{H. Shuhe}, Stochastic Processes Appl. 47, No. 2, 345--352 (1993; Zbl 0786.62066)]. As an application, we give the large deviation inequalities of LS estimator for the simple Michaelis-Menten model.Product matrix processes as limits of random plane partitions.https://www.zbmath.org/1456.600142021-04-16T16:22:00+00:00"Borodin, Alexei"https://www.zbmath.org/authors/?q=ai:borodin.alexei"Gorin, Vadim"https://www.zbmath.org/authors/?q=ai:gorin.vadim"Strahov, Eugene"https://www.zbmath.org/authors/?q=ai:strahov.eugeneSummary: We consider a random process with discrete time formed by squared singular values of products of truncations of Haar-distributed unitary matrices. We show that this process can be understood as a scaling limit of the Schur process, which gives determinantal formulas for (dynamical) correlation functions and a contour integral representation for the correlation kernel. The relation with the Schur processes implies that the continuous limit of marginals for q-distributed plane partitions coincides with the joint law of squared singular values for products of truncations of Haar-distributed random unitary matrices. We provide structural reasons for this coincidence that may also extend to other classes of random matrices.On the convergence and regularity of Aumann-Pettis integrable multivalued martingales.https://www.zbmath.org/1456.600972021-04-16T16:22:00+00:00"El Allali, Mohammed"https://www.zbmath.org/authors/?q=ai:el-allali.mohammed"El-Louh, M'hamed"https://www.zbmath.org/authors/?q=ai:el-louh.mhamed"Ezzaki, Fatima"https://www.zbmath.org/authors/?q=ai:ezzaki.fatimaSummary: We prove a representation of Aumann-Pettis integrable multivalued martingales by Pettis integrable martingale selectors. Regularity of Aumann-Pettis integrable multivalued martingales and their convergence in Mosco sense, Wijsman topology, and linear topology are established.A sufficient and necessary condition of PS-ergodicity of periodic measures and generated ergodic upper expectations.https://www.zbmath.org/1456.600712021-04-16T16:22:00+00:00"Feng, Chunrong"https://www.zbmath.org/authors/?q=ai:feng.chunrong"Qu, Baoyou"https://www.zbmath.org/authors/?q=ai:qu.baoyou"Zhao, Huaizhong"https://www.zbmath.org/authors/?q=ai:zhao.huaizhongLocal asymptotics for the area under the random walk excursion.https://www.zbmath.org/1456.601082021-04-16T16:22:00+00:00"Perfilev, Elena"https://www.zbmath.org/authors/?q=ai:perfilev.elena"Wachtel, Vitali"https://www.zbmath.org/authors/?q=ai:wachtel.vitali-iSummary: We study the tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and light-tailed increments. We determine the asymptotics for local probabilities for the area and prove a local central limit theorem for the duration of the excursion conditioned on the large values of its area.Intermediate-level crossings of a first-passage path.https://www.zbmath.org/1456.602042021-04-16T16:22:00+00:00"Bhat, Uttam"https://www.zbmath.org/authors/?q=ai:bhat.uttam"Redner, S."https://www.zbmath.org/authors/?q=ai:redner.sidneyGeneralized \(k\)-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus.https://www.zbmath.org/1456.600892021-04-16T16:22:00+00:00"Shevchenko, Radomyra"https://www.zbmath.org/authors/?q=ai:shevchenko.radomyra"Slaoui, Meryem"https://www.zbmath.org/authors/?q=ai:slaoui.meryem"Tudor, C. A."https://www.zbmath.org/authors/?q=ai:tudor.ciprian-aSummary: We analyze the generalized \(k\)-variations for the solution to the wave equation driven by an additive Gaussian noise which behaves as a fractional Brownian motion with Hurst parameter \(H \geq \frac{ 1}{ 2}\) in time and which is white in space. The \(k\)-variations are defined along \textit{filters} of any order \(p \geq 1\) and of any length. We show that the sequence of generalized \(k\)-variations satisfies a central limit theorem when \(p > H + \frac{ 1}{ 4}\) and we estimate the rate of convergence for it via the Stein-Malliavin calculus. The results are applied to the estimation of the Hurst index. We construct several consistent estimators for \(H\) and analyze these estimators theoretically and numerically.Hyperuniform and rigid stable matchings.https://www.zbmath.org/1456.601212021-04-16T16:22:00+00:00"Klatt, Michael Andreas"https://www.zbmath.org/authors/?q=ai:klatt.michael-andreas"Last, Günter"https://www.zbmath.org/authors/?q=ai:last.gunter"Yogeshwaran, D."https://www.zbmath.org/authors/?q=ai:yogeshwaran.dhandapaniSummary: We study a stable partial matching \(\tau\) of the \(d\)-dimensional lattice with a stationary determinantal point process \(\Psi\) on \(\mathbb R^d\) with intensity \(\alpha >1\). For instance, \( \Psi\) might be a Poisson process. The matched points from \(\Psi\) form a stationary and ergodic (under lattice shifts) point process \(\Psi^\tau\) with intensity 1 that very much resembles \(\Psi\) for \(\alpha\) close to 1. On the other hand \(\Psi^\tau\) is hyperuniform and number rigid, quite in contrast to a Poisson process. We deduce these properties by proving more general results for a stationary point process \(\Psi \), whose so-called matching flower (a stopping set determining the matching partner of a lattice point) has a certain subexponential tail behavior. For hyperuniformity, we also additionally need to assume some mixing condition on \(\Psi \). Furthermore, if \(\Psi\) is a Poisson process then \(\Psi^\tau\) has an exponentially decreasing truncated pair correlation function.A note on stationary bootstrap variance estimator under long-range dependence.https://www.zbmath.org/1456.620472021-04-16T16:22:00+00:00"Kang, Taegyu"https://www.zbmath.org/authors/?q=ai:kang.taegyu"Kim, Young Min"https://www.zbmath.org/authors/?q=ai:kim.youngmin"Im, Jongho"https://www.zbmath.org/authors/?q=ai:im.jonghoSummary: The stationary bootstrap method is popularly used to compute the standard errors or confidence regions of estimators, generated from time processes exhibiting weakly dependent stationarity. Most previous stationary bootstrap methods have focused on studying large-sample properties of stationary bootstrap inference about a sample mean under short-range dependence. For long-range dependence, recent studies have investigated the properties of block bootstrap methods using overlapping and non-overlapping blocking techniques with fixed block lengths. However, the characteristics of a stationary bootstrap with random block lengths are less well-known under long-range dependence. We investigate the asymptotic property of a stationary bootstrap variance estimator for a sample mean under long-range dependence. Our theoretical and simulation results indicate that the stationary bootstrap method does not have \(\sqrt{ n} \)-consistency for stationary and long-range dependent time processes.High excursions of Gaussian nonstationary processes in discrete time.https://www.zbmath.org/1456.600862021-04-16T16:22:00+00:00"Kozik, I. A."https://www.zbmath.org/authors/?q=ai:kozik.i-a"Piterbarg, V. I."https://www.zbmath.org/authors/?q=ai:piterbarg.vladimir-iSummary: Exact asymptotic behavior is given for high excursion probabilities of Gaussian processes in discrete time as the corresponding lattice pitch unboundedly decreases. The proximity of the asymptotic behavior to that in continuous time is discussed. Examples are given related to fractional Brownian motion and the corresponding ruin problem.Flexible multivariate Hill estimators.https://www.zbmath.org/1456.620922021-04-16T16:22:00+00:00"Dominicy, Yves"https://www.zbmath.org/authors/?q=ai:dominicy.yves"Heikkilä, Matias"https://www.zbmath.org/authors/?q=ai:heikkila.matias"Ilmonen, Pauliina"https://www.zbmath.org/authors/?q=ai:ilmonen.pauliina"Veredas, David"https://www.zbmath.org/authors/?q=ai:veredas.davidSummary: \textit{Y. Dominicy} et al. [``Multivariate Hill estimators'', Int. Stat. Rev. 85, No. 1, 108--142 (2017; \url{doi:10.1111/insr.12120})] introduce a family of Hill estimators for elliptically distributed and heavy tailed random vectors. They propose to use the univariate Hill to a norm of order \(h\) of the data. The norms are homogeneous functions of order one. We show that the family of estimators can be generalized to homogeneous functions of any order and, more importantly, that ellipticity is not required. Only multivariate regular variation is needed, as it is preserved under well-behaved homogeneous functions. This enables us to have flexibility in terms of the estimator and the underlying distribution. Consistency and asymptotic normality are shown, and a Monte Carlo study is conducted to assess the finite sample properties under different asymmetric and heavy tailed multivariate distributions. We illustrate the estimators with an application to 10 years of daily data of paid claims from property insurance policies across 15 regions of Belgium.Filtration shrinkage, the structure of deflators, and failure of market completeness.https://www.zbmath.org/1456.600992021-04-16T16:22:00+00:00"Kardaras, Constantinos"https://www.zbmath.org/authors/?q=ai:kardaras.constantinos"Ruf, Johannes"https://www.zbmath.org/authors/?q=ai:ruf.johannesSummary: We analyse the structure of local martingale deflators projected on smaller filtrations. In a general continuous-path setting, we show that the local martingale parts in the multiplicative Doob-Meyer decomposition of projected local martingale deflators are themselves local martingale deflators in the smaller information market. Via use of a Bayesian filtering approach, we demonstrate the exact mechanism of how updates on the possible class of models under less information result in the strict supermartingale property of projections of such deflators. Finally, we demonstrate that these projections are unable to span all possible local martingale deflators in the smaller information market, by investigating a situation where market completeness is not retained under filtration shrinkage.Data assimilation and parameter estimation for a multiscale stochastic system with \(\alpha \)-stable Lévy noise.https://www.zbmath.org/1456.600932021-04-16T16:22:00+00:00"Zhang, Yanjie"https://www.zbmath.org/authors/?q=ai:zhang.yanjie"Cheng, Zhuan"https://www.zbmath.org/authors/?q=ai:cheng.zhuan"Zhang, Xinyong"https://www.zbmath.org/authors/?q=ai:zhang.xinyong"Chen, Xiaoli"https://www.zbmath.org/authors/?q=ai:chen.xiaoli"Duan, Jinqiao"https://www.zbmath.org/authors/?q=ai:duan.jinqiao"Li, Xiaofan"https://www.zbmath.org/authors/?q=ai:li.xiaofanThe convex hull of the run-and-tumble particle in a plane.https://www.zbmath.org/1456.601052021-04-16T16:22:00+00:00"Hartmann, Alexander K."https://www.zbmath.org/authors/?q=ai:hartmann.alexander-k"Majumdar, Satya N."https://www.zbmath.org/authors/?q=ai:majumdar.satya-n"Schawe, Hendrik"https://www.zbmath.org/authors/?q=ai:schawe.hendrik"Schehr, Grégory"https://www.zbmath.org/authors/?q=ai:schehr.gregoryOn the mean square displacement in Lévy walks.https://www.zbmath.org/1456.601112021-04-16T16:22:00+00:00"Börgers, Christoph"https://www.zbmath.org/authors/?q=ai:borgers.christoph"Greengard, Claude"https://www.zbmath.org/authors/?q=ai:greengard.claudeOn the number of weakly connected subdigraphs in random \(k\)NN digraphs.https://www.zbmath.org/1456.050792021-04-16T16:22:00+00:00"Bahadır, Selim"https://www.zbmath.org/authors/?q=ai:bahadir.selim"Ceyhan, Elvan"https://www.zbmath.org/authors/?q=ai:ceyhan.elvanSummary: We study the number of copies of a weakly connected subdigraph of the \(k\) nearest neighbor \((k\)NN) digraph based on data from certain random point processes in \(\mathbb{R}^d\). In particular, based on the asymptotic theory for functionals of point sets from homogeneous Poisson process (HPP) and uniform binomial process (UBP), we provide a general result for the asymptotic behavior of the number of weakly connected subdigraphs of \(k\) NN digraphs. As corollaries, we obtain asymptotic results for the number of vertices with fixed indegree, the number of shared \(k\)NN pairs, and the number of reflexive \(k\)NNs in the \(k\)NN digraph based on data from HPP and UBP. We also provide several extensions of our results pertaining to the \(k\)NN digraphs; more specifically, the results are extended to the number of weakly connected subdigraphs in a digraph based only on a subset of the first \(k\)NNs, and in a marked or labeled digraph where each vertex also has a mark or a label associated with it, and also to the number of subgraphs of the underlying \(k\)NN graphs. These constructs derived from \(k\)NN digraphs, \(k\)NN graphs, and the marked/labeled \(k\)NN graphs have applications in various fields such as pattern classification and spatial data analysis, and our extensions provide the theoretical basis for certain tools in these areas.Pinned diffusions and Markov bridges.https://www.zbmath.org/1456.600852021-04-16T16:22:00+00:00"Hildebrandt, Florian"https://www.zbmath.org/authors/?q=ai:hildebrandt.florian"Rœlly, Sylvie"https://www.zbmath.org/authors/?q=ai:roelly.sylvieSummary: In this article, we consider a family of real-valued diffusion processes on the time interval [0,1] indexed by their prescribed initial value \(x\in\mathbb{R}\) and another point in space, \(y\in\mathbb{R}\). We first present an \textit{easy-to-check} condition on their drift and diffusion coefficients ensuring that the diffusion is pinned in \(y\) at time \(t=1\). Our main result then concerns the following question: can this family of pinned diffusions be obtained as the bridges either of a Gaussian Markov process or of an Itô diffusion? We eventually illustrate our precise answer with several examples.Well-posedness for Hardy-Hénon parabolic equations with fractional Brownian noise.https://www.zbmath.org/1456.601602021-04-16T16:22:00+00:00"Majdoub, Mohamed"https://www.zbmath.org/authors/?q=ai:majdoub.mohamed"Mliki, Ezzedine"https://www.zbmath.org/authors/?q=ai:mliki.ezzedineThe local solvability of Hardy-Hénon parabolic equations in \(\mathbb{R}^N\) (\(N=2,3\)) perturbed by fractional Brownian noise is discussed in this work. The local pathwise existence and uniqueness of a mild \(\mathbb{L}^q\)-solution under suitable assumptions on \(q\) is established using the contraction mapping principle.
Reviewer: Manil T. Mohan (Roorkee)Lévy noise-induced escape in an excitable system.https://www.zbmath.org/1456.920292021-04-16T16:22:00+00:00"Cai, Rui"https://www.zbmath.org/authors/?q=ai:cai.rui"Chen, Xiaoli"https://www.zbmath.org/authors/?q=ai:chen.xiaoli"Duan, Jinqiao"https://www.zbmath.org/authors/?q=ai:duan.jinqiao"Kurths, Jürgen"https://www.zbmath.org/authors/?q=ai:kurths.jurgen"Li, Xiaofan"https://www.zbmath.org/authors/?q=ai:li.xiaofanDistribution of complex algebraic numbers on the unit circle.https://www.zbmath.org/1456.111852021-04-16T16:22:00+00:00"Götze, F."https://www.zbmath.org/authors/?q=ai:gotze.friedrich-w"Gusakova, A."https://www.zbmath.org/authors/?q=ai:gusakova.anna"Kabluchko, Z."https://www.zbmath.org/authors/?q=ai:kabluchko.zakhar-a"Zaporozhets, D."https://www.zbmath.org/authors/?q=ai:zaporozhets.dmitrySummary: For \(- \pi \leq \beta_1 < \beta_2 \leq \pi \), denote by \(\Phi_{ \beta1,\beta 2}(Q)\) the amount of algebraic numbers of degree \(2m\), elliptic height at most \(Q\), and arguments in \([ \beta_1, \beta_2]\), lying on the unit circle. It is proved that
\[\Phi_{\beta_1,\beta_2}(Q)=Q^{m+1}\int_{\beta_1}^{\beta_2} p(t) dt+O\left(Q^m\log Q\right),\quad Q\to \infty,\]
where \(p(t)\) coincides up to a constant factor with density of the roots of a random trigonometrical polynomial. This density is calculated explicitly using the Edelman-Kostlan formula.Square functions for noncommutative differentially subordinate martingales.https://www.zbmath.org/1456.460522021-04-16T16:22:00+00:00"Jiao, Yong"https://www.zbmath.org/authors/?q=ai:jiao.yong"Randrianantoanina, Narcisse"https://www.zbmath.org/authors/?q=ai:randrianantoanina.narcisse"Wu, Lian"https://www.zbmath.org/authors/?q=ai:wu.lian"Zhou, Dejian"https://www.zbmath.org/authors/?q=ai:zhou.dejianSummary: We prove inequalities involving noncommutative differentially subordinate martingales. More precisely, we prove that if \(x\) is a self-adjoint noncommutative martingale and \(y\) is weakly differentially subordinate to \(x\) then \(y\) admits a decomposition \(dy = a + b + c\) (resp. \( dy = z + w)\) where \(a, b\), and \(c\) are adapted sequences (resp., \(z\) and \(w\) are martingale difference sequences) such that:
\[ \| (a_n)_{n \geq 1} \|_{L_{1, \infty} (\mathcal{M} \overline{\otimes} \ell_{\infty})} + \left\| \left(\sum_{n \geq 1} \mathcal{E}_{n-1} |b_n |^2 \right)^{1/2} \right\|_{1, \infty}
+ \left\| \left(\sum_{n \geq 1} \mathcal{E}_{n-1} |c_n^* |^2 \right)^{1/2} \right\|_{1, \infty} \leq C \| x \|_1
\]
\[
\text{(resp., } \left\| \left(\sum_{n \geq 1} |z_n|^2 \right)^{{1}/{2}}\right\|_{1, \infty}+ \left\| \left(\sum_{n \geq 1} |w_n^*|^2 \right)^{1/2} \right\|_{1, \infty} \leq C \| x \|_1).
\]
We also prove strong-type \((p,p)\) versions of the above weak-type results for \(1 < p < 2\). In order to provide more insights into the interactions between noncommutative differential subordinations and martingale Hardy spaces when \(1 \leq p < 2\), we also provide several martingale inequalities with sharp constants which are new and of independent interest. As a byproduct of our approach, we obtain new and constructive proofs of both the noncommutative Burkholder-Gundy inequalities and the noncommutative Burkholder/Rosenthal inequalities for \(1 < p < 2\) with the optimal order of the constants when \(p \to 1\).Collision of eigenvalues for matrix-valued processes.https://www.zbmath.org/1456.600212021-04-16T16:22:00+00:00"Jaramillo, Arturo"https://www.zbmath.org/authors/?q=ai:jaramillo.arturo"Nualart, David"https://www.zbmath.org/authors/?q=ai:nualart.davidAsymptotics for the systematic and idiosyncratic volatility with large dimensional high-frequency data.https://www.zbmath.org/1456.621722021-04-16T16:22:00+00:00"Kong, Xin-Bing"https://www.zbmath.org/authors/?q=ai:kong.xinbing"Lin, Jin-Guan"https://www.zbmath.org/authors/?q=ai:lin.jinguan"Liu, Guang-Ying"https://www.zbmath.org/authors/?q=ai:liu.guangyingA comparison principle for random walk on dynamical percolation.https://www.zbmath.org/1456.602562021-04-16T16:22:00+00:00"Hermon, Jonathan"https://www.zbmath.org/authors/?q=ai:hermon.jonathan"Sousi, Perla"https://www.zbmath.org/authors/?q=ai:sousi.perlaSummary: We consider the model of random walk on dynamical percolation introduced by \textit{Y. Peres} et al. [Probab. Theory Relat. Fields 162, No. 3--4, 487--530 (2015; Zbl 1326.60140)]. We obtain comparison results for this model for hitting and mixing times and for the spectral gap and log-Sobolev constant with the corresponding quantities for simple random walk on the underlying graph \(G\), for general graphs. When \(G\) is the torus \(\mathbb{Z}_n^d\), we recover the results of Peres et al. [loc. cit.], and we also extend them to the critical case. We also obtain bounds in the cases where \(G\) is a transitive graph of moderate growth and also when it is the hypercube.Rare events in stochastic processes with sub-exponential distributions and the big jump principle.https://www.zbmath.org/1456.601242021-04-16T16:22:00+00:00"Burioni, Raffaella"https://www.zbmath.org/authors/?q=ai:burioni.raffaella"Vezzani, Alessandro"https://www.zbmath.org/authors/?q=ai:vezzani.alessandroAttractiveness and exponential \(p\)-stability of neutral stochastic functional integro-differential equations driven by Wiener process and fBm with impulses effects.https://www.zbmath.org/1456.601732021-04-16T16:22:00+00:00"Hamit, Mahamat Hassan Mahamat"https://www.zbmath.org/authors/?q=ai:hamit.mahamat-hassan-mahamat"Allognissode, Fulbert Kuessi"https://www.zbmath.org/authors/?q=ai:allognissode.fulbert-kuessi"Mohamed, Mohamed salem"https://www.zbmath.org/authors/?q=ai:mohamed.mohamed-salem"Issaka, Louk-Man"https://www.zbmath.org/authors/?q=ai:issaka.louk-man"Diop, Mamadou Abdoul"https://www.zbmath.org/authors/?q=ai:diop.mamadou-abdoulSummary: In this work, we consider a class of neutral stochastic integro-differential equations driven by Wiener process and fractional Brownian motion with impulses effects. This paper deals with the global attractiveness and quasi-invariant sets for neutral stochastic integro-differential equations driven by Wiener process and fractional Brownian motion with impulses effects in Hilbert spaces. We use new integral inequalities combined with theories of resolvent operators to establish a set of sufficient conditions for the exponential \(p\)-stability of the mild solution of the considered equations. An example is presented to demonstrate the obtained theory.Localization in Gaussian disordered systems at low temperature.https://www.zbmath.org/1456.602702021-04-16T16:22:00+00:00"Bates, Erik"https://www.zbmath.org/authors/?q=ai:bates.erik"Chatterjee, Sourav"https://www.zbmath.org/authors/?q=ai:chatterjee.sourav.1Summary: For a broad class of Gaussian disordered systems at low temperature, we show that the Gibbs measure is asymptotically localized in small neighborhoods of a small number of states. From a single argument, we obtain: (i) a version of ``complete'' path localization for directed polymers that is not available even for exactly solvable models, and (ii) a result about the exhaustiveness of Gibbs states in spin glasses not requiring the Ghirlanda-Guerra identities.Unbiased truncated quadratic variation for volatility estimation in jump diffusion processes.https://www.zbmath.org/1456.620382021-04-16T16:22:00+00:00"Amorino, Chiara"https://www.zbmath.org/authors/?q=ai:amorino.chiara"Gloter, Arnaud"https://www.zbmath.org/authors/?q=ai:gloter.arnaudSummary: The problem of integrated volatility estimation for an Ito semimartingale is considered under discrete high-frequency observations in short time horizon. We provide an asymptotic expansion for the integrated volatility that gives us, in detail, the contribution deriving from the jump part. The knowledge of such a contribution allows us to build an unbiased version of the truncated quadratic variation, in which the bias is visibly reduced. In earlier results to have the original truncated realized volatility well-performed the condition \(\beta>\frac{1}{2(2-\alpha)}\) on \(\beta\) (that is such that \((\frac{1}{n})^\beta\) is the threshold of the truncated quadratic variation) and on the degree of jump activity \(\alpha\) was needed (see [\textit{C. Mancini}, ibid. 121, No. 4, 845--855 (2011; Zbl 1216.62159); \textit{J. Jacod}, ibid. 118, No. 4, 517--559 (2008; Zbl 1142.60022)]). In this paper we theoretically relax this condition and we show that our unbiased estimator achieves excellent numerical results for any couple \((\alpha,\beta)\).The tempered stable process with infinitely divisible inverse subordinators.https://www.zbmath.org/1456.828272021-04-16T16:22:00+00:00"Wyłomańska, Agnieszka"https://www.zbmath.org/authors/?q=ai:wylomanska.agnieszkaA note on the cross-covariance operator and on congruence relations for Hilbert space valued stochastic processes.https://www.zbmath.org/1456.621112021-04-16T16:22:00+00:00"King, David"https://www.zbmath.org/authors/?q=ai:king.david-a|king.david-jSummary: Explicit formulas are derived for the congruence mappings that connect three Hilbert spaces associated with a second-order stochastic process. In particular, an insightful expression is obtained for the mapping that connects a process to its corresponding reproducing kernel Hilbert space. In addition, a useful infinite dimensional extension of a result from \textit{C. G. Khatri} [Gujarat Stat. Rev. 3, No. 2, 21--23 (1976; Zbl 0341.62015)] which pertains to cross-covariance operators is provided.Observation time dependent mean first passage time of diffusion and subdiffusion processes.https://www.zbmath.org/1456.602182021-04-16T16:22:00+00:00"Kim, Ji-Hyun"https://www.zbmath.org/authors/?q=ai:kim.jihyun|kim.ji-hyun"Lee, Hunki"https://www.zbmath.org/authors/?q=ai:lee.hunki"Song, Sanggeun"https://www.zbmath.org/authors/?q=ai:song.sanggeun"Koh, Hye Ran"https://www.zbmath.org/authors/?q=ai:koh.hye-ran"Sung, Jaeyoung"https://www.zbmath.org/authors/?q=ai:sung.jaeyoungSpatio-temporal dependence measures for bivariate AR(1) models with \(\alpha \)-stable noise.https://www.zbmath.org/1456.621902021-04-16T16:22:00+00:00"Grzesiek, Aleksandra"https://www.zbmath.org/authors/?q=ai:grzesiek.aleksandra"Sikora, Grzegorz"https://www.zbmath.org/authors/?q=ai:sikora.grzegorz"Teuerle, Marek"https://www.zbmath.org/authors/?q=ai:teuerle.marek-a"Wyłomańska, Agnieszka"https://www.zbmath.org/authors/?q=ai:wylomanska.agnieszkaThe authors investigate properties of the \(\alpha\)-stable bidimensional
vector autoregressive VAR(1) model described by the equation
\[
X (t) -\Theta X (t - 1) = Z (t) ,
\] where the noise \(\{Z (t)\}\)
is an \(\alpha\)-stable vector in \(\mathbb R^2\) with the stability index \(\alpha<2\)
called \(\alpha\)-stable noise (or an infinite-variance noise).
Under the condition that all the eigenvalues of the matrix \(\Theta\) are less than 1 in absolute value, which is equivalent to
\(\det(I - z\Theta)\not= 0\) for all \(\{z:|z|\leq 1\}\),
the defined by such an equation time series \(\{X (t)\}\) can be written in causal representation
\[
X(t) =\sum_{j=0}^{\infty}(\Theta)^jZ (t - j)
\]
In the case of (Gaussian) white noise \(\{Z (t)\}\) the spatio-temporal dependence structure of the bidimensional time series the cross-covariation is applied to describe properties of the time series. The cross-covariance has found many applications in time series investigation, especially in signal processing.
However, the cross-covariance is not an appropriate measure for the \(\alpha\)-stable stochastic processes where the second moment is infinite and therefore the theoretical function does not exist.
Then, for stochastic processes with infinite variance, the alternative measures should be applied.
In this article the authors propose the cross-codifference and the cross-covariation functions which are the analogues of the classical cross-covariance for infinite variance processes.
They provide theoretical results for cross-codifference and cross-covariation bidimensional VAR(1) time series with \(\alpha\)-stable i.i.d. noise and demonstrate that cross-codifference and cross-covariation can give different useful information about the relationships between components of bidimensional VAR models.
This article is an extension of the authors previous work (see [``Cross-codifference for bidimensional VAR(1) time series with infinite variance'', Comm. Stat. (to appear)]) where the cross-codifference was considered as the spatio-temporal measure of the components of VAR model based on sub-Gaussian distribution.
Reviewer: Mikhail P. Moklyachuk (Kyïv)Deterministic walks in random environment.https://www.zbmath.org/1456.602692021-04-16T16:22:00+00:00"Aimino, Romain"https://www.zbmath.org/authors/?q=ai:aimino.romain"Liverani, Carlangelo"https://www.zbmath.org/authors/?q=ai:liverani.carlangeloSummary: Motivated by the random Lorentz gas, we study deterministic walks in random environment and show that (in simple, yet relevant cases) they can be reduced to a class of random walks in random environment where the jump probability depends (weakly) on the past. In addition, we prove few basic results (hopefully, the germ of a general theory) for the latter purely probabilistic model.On Baxter type theorems for generalized random Gaussian processes with independent values.https://www.zbmath.org/1456.600872021-04-16T16:22:00+00:00"Krasnitskiy, S. M."https://www.zbmath.org/authors/?q=ai:krasnitskii.sergei-m"Kurchenko, O. O."https://www.zbmath.org/authors/?q=ai:kurchenko.oleksandr-oSummary: We construct suitable families of basic functions and prove theorems of Baxter type for generalized Gaussian random processes with independent values. These theorems are used to divide families of such processes into classes. The singularity of probability measures corresponding to representatives of different classes is proved.Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory.https://www.zbmath.org/1456.820792021-04-16T16:22:00+00:00"Torquato, Salvatore"https://www.zbmath.org/authors/?q=ai:torquato.salvatore"Scardicchio, A."https://www.zbmath.org/authors/?q=ai:scardicchio.antonello"Zachary, Chase E."https://www.zbmath.org/authors/?q=ai:zachary.chase-eStochastic acceleration in generalized squared Bessel processes.https://www.zbmath.org/1456.601692021-04-16T16:22:00+00:00"Valenti, D."https://www.zbmath.org/authors/?q=ai:valenti.davide"Chichigina, O. A."https://www.zbmath.org/authors/?q=ai:chichigina.olga-a"Dubkov, A. A."https://www.zbmath.org/authors/?q=ai:dubkov.alexander-a"Spagnolo, B."https://www.zbmath.org/authors/?q=ai:spagnolo.bernardoPhase transition in a random minima model: mean field theory and exact solution on the Bethe lattice.https://www.zbmath.org/1456.823862021-04-16T16:22:00+00:00"Sollich, Peter"https://www.zbmath.org/authors/?q=ai:sollich.peter"Majumdar, Satya N."https://www.zbmath.org/authors/?q=ai:majumdar.satya-n"Bray, Alan J."https://www.zbmath.org/authors/?q=ai:bray.alan-jAn \(L^p\) multiplicative coboundary theorem for sequences of unitriangular random matrices.https://www.zbmath.org/1456.600242021-04-16T16:22:00+00:00"Morrow, Steven T."https://www.zbmath.org/authors/?q=ai:morrow.steven-tSummary: \textit{R. C. Bradley} [ibid.. 9, No. 3, 659--678 (1996; Zbl 0870.60028)] proved a ``multiplicative coboundary'' theorem for sequences of unitriangular random matrices with integer entries, requiring tightness of the family of distributions of the entries from the partial matrix products of the sequence. This was an analog of \textit{K. Schmidt}'s result [Cocycles on ergodic transformation groups. Macmillan Lectures in Mathematics 1. Delhi, Bombay, Calcutta, Madras: The Macmillan Company of India Ltd. (1977; Zbl 0421.28017)] for sequences of real-valued random variables with tightness of the family of partial sums. Here is an \(L^p\) moment analog of Bradley's result which also relaxes the restriction of entries being integers.Record statistics for a discrete-time random walk with correlated steps.https://www.zbmath.org/1456.601062021-04-16T16:22:00+00:00"Kearney, Michael J."https://www.zbmath.org/authors/?q=ai:kearney.michael-jLight scattering as a Poisson process and first-passage probability.https://www.zbmath.org/1456.602722021-04-16T16:22:00+00:00"Zeller, Claude"https://www.zbmath.org/authors/?q=ai:zeller.claude"Cordery, Robert"https://www.zbmath.org/authors/?q=ai:cordery.robertBoundary behavior of random walks in cones.https://www.zbmath.org/1456.601102021-04-16T16:22:00+00:00"Raschel, Kilian"https://www.zbmath.org/authors/?q=ai:raschel.kilian"Tarrago, Pierre"https://www.zbmath.org/authors/?q=ai:tarrago.pierreSummary: We study the asymptotic behavior of zero-drift random walks confined to multidimensional convex cones, when the endpoint is close to the boundary. We derive a local limit theorem in the fluctuation regime.Global properties of stochastic Loewner evolution driven by Lévy processes.https://www.zbmath.org/1456.827832021-04-16T16:22:00+00:00"Oikonomou, P."https://www.zbmath.org/authors/?q=ai:oikonomou.p"Rushkin, I."https://www.zbmath.org/authors/?q=ai:rushkin.i"Gruzberg, I. A."https://www.zbmath.org/authors/?q=ai:gruzberg.ilya-a"Kadanoff, L. P."https://www.zbmath.org/authors/?q=ai:kadanoff.leo-pMarkovian dynamics of exchangeable arrays.https://www.zbmath.org/1456.600762021-04-16T16:22:00+00:00"Černý, Jiří"https://www.zbmath.org/authors/?q=ai:cerny.jiri"Klimovsky, Anton"https://www.zbmath.org/authors/?q=ai:klimovsky.antonA growth-fragmentation model related to Ornstein-Uhlenbeck type processes.https://www.zbmath.org/1456.601172021-04-16T16:22:00+00:00"Shi, Quan"https://www.zbmath.org/authors/?q=ai:shi.quanSummary: Growth-fragmentation processes describe systems of particles in which each particle may grow larger or smaller, and divide into smaller ones as time proceeds. Unlike previous studies, which have focused mainly on the self-similar case, we introduce a new type of growth-fragmentation which is closely related to Lévy driven Ornstein-Uhlenbeck type processes. Our model can be viewed as a generalization of compensated fragmentation processes introduced by Bertoin, or the stochastic counterpart of a family of growth-fragmentation equations. We establish a convergence criterion for a sequence of such growth-fragmentations. We also prove that, under certain conditions, this system fulfills a law of large numbers.Non-Gaussian features of chaotic Hamiltonian transport.https://www.zbmath.org/1456.370392021-04-16T16:22:00+00:00"Venegeroles, Roberto"https://www.zbmath.org/authors/?q=ai:venegeroles.roberto"Saa, Alberto"https://www.zbmath.org/authors/?q=ai:saa.albertoDynamical robustness of discrete conservative systems: Harper and generalized standard maps.https://www.zbmath.org/1456.601302021-04-16T16:22:00+00:00"Tirnakli, Ugur"https://www.zbmath.org/authors/?q=ai:tirnakli.ugur"Tsallis, Constantino"https://www.zbmath.org/authors/?q=ai:tsallis.constantino"Cetin, Kivanc"https://www.zbmath.org/authors/?q=ai:cetin.kivancOn multidimensional record patterns.https://www.zbmath.org/1456.601262021-04-16T16:22:00+00:00"Krapivsky, P. L."https://www.zbmath.org/authors/?q=ai:krapivsky.pavel-l"Luck, J. M."https://www.zbmath.org/authors/?q=ai:luck.jean-marcConnectivity properties of the adjacency graph of \(\text{SLE}_{\kappa}\) bubbles for \(\kappa\in(4,8)\).https://www.zbmath.org/1456.602142021-04-16T16:22:00+00:00"Gwynne, Ewain"https://www.zbmath.org/authors/?q=ai:gwynne.ewain"Pfeffer, Joshua"https://www.zbmath.org/authors/?q=ai:pfeffer.joshuaSummary: We study the adjacency graph of bubbles, that is, complementary connected components of a \(\text{SLE}_{\kappa}\) curve for \(\kappa\in (4,8)\), with two such bubbles considered to be adjacent if their boundaries intersect. We show that this adjacency graph is a.s. connected for \(\kappa\in (4,\kappa_0]\), where \(\kappa_0\approx 5.6158\) is defined explicitly. This gives a partial answer to a problem posed by \textit{B. Duplantier} et al. [``Liouville quantum gravity as a mating of trees'', Preprint, \url{arXiv:1409.7055}]. Our proof in fact yields a stronger connectivity result for \(\kappa\in (4,\kappa_0]\), which says that there is a Markovian way of finding a path from any fixed bubble to \(\infty \). We also show that there is a (nonexplicit) \(\kappa_1\in (\kappa_0,8)\) such that this stronger condition does not hold for \(\kappa\in [\kappa_1,8)\).
Our proofs are based on an encoding of \(\text{SLE}_{\kappa}\) in terms of a pair of independent \(\kappa/4\)-stable processes, which allows us to reduce our problem to a problem about stable processes. In fact, due to this encoding, our results can be rephrased as statements about the connectivity of the adjacency graph of loops when one glues together an independent pair of so-called \(\kappa/4\)-stable looptrees, as studied, for example, by \textit{N. Curien} and \textit{I. Kortchemski} [Electron. J. Probab. 19, Paper No. 108, 35 p. (2014; Zbl 1307.60061)]
The above encoding comes from the theory of Liouville quantum gravity (LQG), but the paper can be read without any knowledge of LQG if one takes the encoding as a black box.Book review of: G. Aubrun and S. Szarek, Alice and Bob meet Banach: the interface of asymptotic geometric analysis and quantum information theory.https://www.zbmath.org/1456.000092021-04-16T16:22:00+00:00"Brannan, Michael"https://www.zbmath.org/authors/?q=ai:brannan.michaelReview of [Zbl 1402.46001].Entropic repulsion for the occupation-time field of random interlacements conditioned on disconnection.https://www.zbmath.org/1456.601892021-04-16T16:22:00+00:00"Chiarini, Alberto"https://www.zbmath.org/authors/?q=ai:chiarini.alberto"Nitzschner, Maximilian"https://www.zbmath.org/authors/?q=ai:nitzschner.maximilianSummary: We investigate percolation of the vacant set of random interlacements on \(\mathbb{Z}^d,d\geq 3\), in the strongly percolative regime. We consider the event that the interlacement set at level \(u\) disconnects the discrete blow-up of a compact set \(A\subseteq\mathbb{R}^d\) from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the occupation times deviate from a specific function depending on the harmonic potential of \(A\), when disconnection occurs. If certain critical levels coincide, which is plausible but open at the moment, these bounds imply that conditionally on disconnection, the occupation-time profile undergoes an entropic push governed by a specific function depending on \(A\). Similar entropic repulsion phenomena conditioned on disconnection by level-sets of the discrete Gaussian free field on \(\mathbb{Z}^d,d\geq 3\), have been obtained by the authors in [Probab. Theory Relat. Fields 177, No. 1--2, 525--575 (2020; Zbl 07202717)]. Our proofs rely crucially on the ``solidification estimates'' developed in [the second author and \textit{A.-S. Sznitman}, J. Eur. Math. Soc. (JEMS) 22, No. 8, 2629--2672 (2020; Zbl 07227743)].Distribution of extreme first passage times of diffusion.https://www.zbmath.org/1456.920072021-04-16T16:22:00+00:00"Lawley, Sean D."https://www.zbmath.org/authors/?q=ai:lawley.sean-dSummary: Many events in biology are triggered when a diffusing searcher finds a target, which is called a first passage time (FPT). The overwhelming majority of FPT studies have analyzed the time it takes a single searcher to find a target. However, the more relevant timescale in many biological systems is the time it takes the fastest searcher(s) out of many searchers to find a target, which is called an extreme FPT. In this paper, we apply extreme value theory to find a tractable approximation for the full probability distribution of extreme FPTs of diffusion. This approximation can be easily applied in many diverse scenarios, as it depends on only a few properties of the short time behavior of the survival probability of a single FPT. We find this distribution by proving that a careful rescaling of extreme FPTs converges in distribution as the number of searchers grows. This limiting distribution is a type of Gumbel distribution and involves the LambertW function. This analysis yields new explicit formulas for approximations of statistics of extreme FPTs (mean, variance, moments, etc.) which are highly accurate and are accompanied by rigorous error estimates.On single-layer potentials for a class of pseudo-differential equations related to linear transformations of a symmetric \(\alpha \)-stable stochastic process.https://www.zbmath.org/1456.601202021-04-16T16:22:00+00:00"Mamalyha, Kh. V."https://www.zbmath.org/authors/?q=ai:mamalyha.kh-v"Osypchuk, M. M."https://www.zbmath.org/authors/?q=ai:osypchuk.m-mSummary: In this article an arbitrary invertible linear transformations of a symmetric \(\alpha \)-stable stochastic process in \(d\)-dimensional Euclidean space \(\mathbb{R}^d\) are investigated. The result of such transformation is a Markov process in \(\mathbb{R}^d\) whose generator is the pseudo-differential operator defined by its symbol \((-(Q\xi,\xi)^{\alpha/2})_{\xi\in\mathbb{R}^d}\) with some symmetric positive definite \(d\times d\)-matrix \(Q\) and fixed exponent \(\alpha\in(1,2)\). The transition probability density of this process is the fundamental solution of some parabolic pseudo-differential equation. The notion of a single-layer potential for that equation is introduced and its properties are investigated. In particular, an operator is constructed whose role in our consideration is analogous to that the gradient in the classical theory. An analogy to the classical theorem on the jump of the co-normal derivative of the single-layer potential is proved. This result can be applied for solving some boundary-value problems for the parabolic pseudo-differential equations under consideration. For \(\alpha = 2 \), the process under consideration is a linear transformation of Brownian motion, and all the investigated properties of the single-layer potential are well known.The identification problem for BSDEs driven by possibly non-quasi-left-continuous random measures.https://www.zbmath.org/1456.602242021-04-16T16:22:00+00:00"Bandini, Elena"https://www.zbmath.org/authors/?q=ai:bandini.elena"Russo, Francesco"https://www.zbmath.org/authors/?q=ai:russo.francesco.2Multi-condition of stability for nonlinear stochastic non-autonomous delay differential equation.https://www.zbmath.org/1456.340782021-04-16T16:22:00+00:00"Shaikhet, Leonid"https://www.zbmath.org/authors/?q=ai:shaikhet.leonid-eSummary: A nonlinear stochastic differential equation with the order of nonlinearity higher than one, with several discrete and distributed delays and time varying coefficients is considered. It is shown that the sufficient conditions for exponential mean square stability of the linear part of the considered nonlinear equation also are sufficient conditions for stability in probability of the initial nonlinear equation. Some new sufficient condition of stability in probability for the zero solution of the considered nonlinear non-autonomous stochastic differential equation is obtained which can be considered as a multi-condition of stability because it allows to get for one considered equation at once several different complementary of each other sufficient stability conditions. The obtained results are illustrated with numerical simulations and figures.Bridges with random length: gamma case.https://www.zbmath.org/1456.600952021-04-16T16:22:00+00:00"Erraoui, Mohamed"https://www.zbmath.org/authors/?q=ai:erraoui.mohamed"Hilbert, Astrid"https://www.zbmath.org/authors/?q=ai:hilbert.astrid"Louriki, Mohammed"https://www.zbmath.org/authors/?q=ai:louriki.mohammedSummary: In this paper, we generalize the concept of gamma bridge in the sense that the length will be random, that is, the time to reach the given level is random. The main objective of this paper is to show that certain basic properties of gamma bridges with deterministic length stay true also for gamma bridges with random length. We show that the gamma bridge with random length is a pure jump process and that its jumping times are countable and dense in the random interval bounded by 0 and the random length. Moreover, we prove that this process is a Markov process with respect to its completed natural filtration as well as with respect to the usual augmentation of this filtration, which leads us to conclude that its completed natural filtration is right continuous. Finally, we give its canonical decomposition with respect to the usual augmentation of its natural filtration.Extracting non-Gaussian governing laws from data on mean exit time.https://www.zbmath.org/1456.370932021-04-16T16:22:00+00:00"Zhang, Yanxia"https://www.zbmath.org/authors/?q=ai:zhang.yanxia"Duan, Jinqiao"https://www.zbmath.org/authors/?q=ai:duan.jinqiao"Jin, Yanfei"https://www.zbmath.org/authors/?q=ai:jin.yanfei"Li, Yang"https://www.zbmath.org/authors/?q=ai:li.yang.5Summary: Motivated by the existing difficulties in establishing mathematical models and in observing state time series for some complex systems, especially for those driven by non-Gaussian Lévy motion, we devise a method for extracting non-Gaussian governing laws with observations only on the mean exit time. It is feasible to observe the mean exit time for certain complex systems. With such observations, we use a sparse regression technique in the least squares sense to obtain the approximated function expression of the mean exit time. Then, we learn the generator and further identify the governing stochastic differential equation by solving an inverse problem for a nonlocal partial differential equation and minimizing an error objective function. Finally, we verify the efficacy of the proposed method by three examples with the aid of the simulated data from the original systems. Results show that our method can apply to not only the stochastic dynamical systems driven by Gaussian Brownian motion but also those driven by non-Gaussian Lévy motion, including those systems with complex rational drift.
{\copyright 2020 American Institute of Physics}Response behavior of aging systems with temporal disorder.https://www.zbmath.org/1456.602732021-04-16T16:22:00+00:00"Eule, Stephan"https://www.zbmath.org/authors/?q=ai:eule.stephanOn hyperbolic decay of prediction error variance for deterministic stationary sequences.https://www.zbmath.org/1456.600772021-04-16T16:22:00+00:00"Babayan, N. M."https://www.zbmath.org/authors/?q=ai:babayan.n-m.1"Ginovyan, M. S."https://www.zbmath.org/authors/?q=ai:ginovyan.mamikon-sSummary: One of the main problems in prediction theory of second-order stationary processes, called direct prediction problem, is to describe the asymptotic behavior of the best linear mean squared one-step ahead prediction error variance in predicting the value \(X(0)\) of a stationary process \(X(t)\) by the observed past of finite length \(n\) as \(n\) goes to infinity, depending on the regularity nature (deterministic or non-deterministic) of the underlying observed process \(X(t)\). In this paper, we obtain sufficient conditions for hyperbolic decay of prediction error variance for deterministic stationary sequences, generalizing a result obtained by
\textit{M. Rosenblatt} [J. Math. Mech. 6, 801--810 (1957; Zbl 0080.35001)].Duals of random vectors and processes with applications to prediction problems with missing values.https://www.zbmath.org/1456.622262021-04-16T16:22:00+00:00"Kasahara, Yukio"https://www.zbmath.org/authors/?q=ai:kasahara.yukio"Pourahmadi, Mohsen"https://www.zbmath.org/authors/?q=ai:pourahmadi.mohsen"Inoue, Akihiko"https://www.zbmath.org/authors/?q=ai:inoue.akihikoSummary: Important results in prediction theory dealing with missing values have been obtained traditionally using difficult techniques based on duality in Hilbert spaces of analytic functions [\textit{T. Nakazi}, Stud. Math. 78, 7--14 (1984; Zbl 0608.60040); \textit{A. G. Miamee} and \textit{M. Pourahmadi}, J. Lond. Math. Soc., II. Ser. 38, No. 1, 133--145 (1988; Zbl 0669.41018)]. We obtain and unify these results using a simple finite-dimensional duality lemma which is essentially an abstraction of a regression property of a multivariate normal random vector [\textit{M. M. Rao}, Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B 8, 319--332 (1973; Zbl 0251.60034)] or its inverse covariance matrix. The approach reveals the roles of duality and biorthogonality of random vectors in dealing with infinite-dimensional and difficult prediction problems. A novelty of this approach is its reliance on the explicit representation of the prediction error in terms of the data rather than the predictor itself as in the traditional techniques. In particular, we find a new and explicit formula for the dual of the semi-finite process \(\{X_t;t\leq n\}\) for a fixed \(n\), which does not seem to be possible using the existing techniques.Double roots of random Littlewood polynomials.https://www.zbmath.org/1456.601292021-04-16T16:22:00+00:00"Peled, Ron"https://www.zbmath.org/authors/?q=ai:peled.ron"Sen, Arnab"https://www.zbmath.org/authors/?q=ai:sen.arnab"Zeitouni, Ofer"https://www.zbmath.org/authors/?q=ai:zeitouni.oferSummary: We consider random polynomials whose coefficients are independent and uniform on \(\{-1,1\}\). We prove that the probability that such a polynomial of degree \(n\) has a double root is \(o(n^{-2})\) when \(n+1\) is not divisible by 4 and asymptotic to \(\frac{8\sqrt 3}{\pi n^2}\) otherwise. This result is a corollary of a more general theorem that we prove concerning random polynomials with independent, identically distributed coefficients having a distribution which is supported on \(\{-1,0,1\}\) and whose largest atom is strictly less than \(1/\sqrt 3\). In this general case, we prove that the probability of having a double root equals the probability that either \(-1\), \(0\) or \(1\) are double roots up to an \(o(n^{-2})\) factor and we find the asymptotics of the latter probability.Stopping with expectation constraints: 3 points suffice.https://www.zbmath.org/1456.600942021-04-16T16:22:00+00:00"Ankirchner, Stefan"https://www.zbmath.org/authors/?q=ai:ankirchner.stefan"Kazi-Tani, Nabil"https://www.zbmath.org/authors/?q=ai:kazi-tani.nabil"Klein, Maike"https://www.zbmath.org/authors/?q=ai:klein.maike"Kruse, Thomas"https://www.zbmath.org/authors/?q=ai:kruse.thomasThe paper deals with a problem of optimally stopping a process with a stopping time satisfying an expectation constraint. Let \((Y_t)_{t\in\Re^+}\) be a one-dimensional regular continuous strong Markov process with respect to a right-continuous filtration \((\mathcal{F}_t)\). The state space \(J\subset\Re\) is assumed to be an open, half-open or closed interval. The payoff \(f : \Re\rightarrow\Re\) is a Borel-measurable having regularity conditions. Let us denote by \(\mathcal{T} (\mathbf{T})\) the set of \((\mathcal{F}_t)\)-stopping times such that \(\mathbf{E}[\tau]\leq \mathbf{T}\in \Re^+\). The problem considered in the paper has the form: \(\max_{\tau\in \mathcal{T} (\mathbf{T})} \textbf{E}[f(Y_\tau)]\).
It is shown that to obtain a solution of such constrained optimization problem it suffices to consider only stopping times such that the law of the process at the stopping time is a weighted sum of 3 Dirac measures. The idea for proving a reduction to 3 Dirac measures is based on rewriting the considered constrained stopping problem as a linear optimization problem over a set of probability measures. Recent results of \textit{S. Ankirchner} et al. [Bernoulli 21, No. 2, 1067--1088 (2015; Zbl 1328.60101)] and \textit{D. Hobson} [Electron. J. Probab. 20, Paper No. 83, 26 p. (2015; Zbl 1328.60104)] on the Skorokhod embedding problem characterizing the set \(\mathcal{A}(\mathbf{T})\) of probability distributions that can be embedded into \(Y\) with stopping times having expectation smaller than or equal to \(\mathbf{T}\). As for standard linear problems the maximal value of the optimization is attained by extreme points. The extreme points of \(\mathcal{A}(\mathbf{T})\) turn out to be contained in the set of probability measures that can be written as weighted sums of at most 3 Dirac measures.
Reviewer: Krzysztof J. Szajowski (Wrocław)Bayesian model selection with fractional Brownian motion.https://www.zbmath.org/1456.620442021-04-16T16:22:00+00:00"Krog, Jens"https://www.zbmath.org/authors/?q=ai:krog.jens"Jacobsen, Lars H."https://www.zbmath.org/authors/?q=ai:jacobsen.lars-h"Lund, Frederik W."https://www.zbmath.org/authors/?q=ai:lund.frederik-w"Wüstner, Daniel"https://www.zbmath.org/authors/?q=ai:wustner.daniel"Lomholt, Michael A."https://www.zbmath.org/authors/?q=ai:lomholt.michael-aPower of the MOSUM test for online detection of a transient change in mean.https://www.zbmath.org/1456.601072021-04-16T16:22:00+00:00"Noonan, Jack"https://www.zbmath.org/authors/?q=ai:noonan.jack"Zhigljavsky, Anatoly"https://www.zbmath.org/authors/?q=ai:zhigljavsky.anatoly-aSummary: In this article we discuss an online moving sum (MOSUM) test for detection of a transient change in the mean of a sequence of independent and identically distributed (i.i.d.) normal random variables. By using a well-developed theory for continuous time Gaussian processes and subsequently correcting the results for discrete time, we provide accurate approximations for the average run length (ARL) and power of the test. We check theoretical results against simulations, compare the power of the MOSUM test with that of the cumulative sum (CUSUM), and briefly consider the cases of nonnormal random variables and weighted sums.Resonant activation in 2D and 3D systems driven by multi-variate Lévy noise.https://www.zbmath.org/1456.600922021-04-16T16:22:00+00:00"Szczepaniec, Krzysztof"https://www.zbmath.org/authors/?q=ai:szczepaniec.krzysztof"Dybiec, Bartłomiej"https://www.zbmath.org/authors/?q=ai:dybiec.bartlomiejKingman's coalescent with erosion.https://www.zbmath.org/1456.602312021-04-16T16:22:00+00:00"Foutel-Rodier, Félix"https://www.zbmath.org/authors/?q=ai:foutel-rodier.felix"Lambert, Amaury"https://www.zbmath.org/authors/?q=ai:lambert.amaury"Schertzer, Emmanuel"https://www.zbmath.org/authors/?q=ai:schertzer.emmanuelSummary: Consider the Markov process taking values in the partitions of \(\mathbb{N}\) such that each pair of blocks merges at rate one, and each integer is eroded, i.e., becomes a singleton block, at rate \(d\). This is a special case of exchangeable fragmentation-coalescence process, called Kingman's coalescent with erosion. We provide a new construction of the stationary distribution of this process as a sample from a standard flow of bridges. This allows us to give a representation of the asymptotic frequencies of this stationary distribution in terms of a sequence of independent diffusions. Moreover, we introduce a new process valued in the partitions of \(\mathbb{Z}\) called Kingman's coalescent with immigration, where pairs of blocks coalesce at rate one, and new blocks of size one immigrate according to a Poisson process of intensity \(d\). By coupling Kingman's coalescents with erosion and with immigration, we are able to show that the size of a block chosen uniformly at random from the stationary distribution of the restriction of Kingman's coalescent with erosion to \(\{1,\dots,n\}\) converges as \(n\to\infty\) to the total progeny of a critical binary branching process.A local stable bootstrap for power variations of pure-jump semimartingales and activity index estimation.https://www.zbmath.org/1456.621712021-04-16T16:22:00+00:00"Hounyo, Ulrich"https://www.zbmath.org/authors/?q=ai:hounyo.ulrich"Varneskov, Rasmus T."https://www.zbmath.org/authors/?q=ai:varneskov.rasmus-tSummary: We provide a new resampling procedure-the local stable bootstrap-that is able to mimic the dependence properties of realized power variations for pure-jump semimartingales observed at different frequencies. This allows us to propose a bootstrap estimator and inference procedure for the activity index of the underlying process, \(\beta\), as well as bootstrap tests for whether it obeys a jump-diffusion or a pure-jump process, that is, of the null hypothesis \(\mathcal{H}_0:\beta=2\) against the alternative \(\mathcal{H}_1:\beta<2\). We establish first-order asymptotic validity of the resulting bootstrap power variations, activity index estimator, and diffusion tests for \(\mathcal{H}_0\). Moreover, the finite sample size and power properties of the proposed diffusion tests are compared to those of benchmark tests using Monte Carlo simulations. Unlike existing procedures, our bootstrap tests are correctly sized in general settings. Finally, we illustrate the use and properties of the new bootstrap diffusion tests using high-frequency data on three FX series, the S\&P 500, and the VIX.On the probability that a stationary Gaussian process with spectral gap remains non-negative on a long interval.https://www.zbmath.org/1456.600782021-04-16T16:22:00+00:00"Feldheim, Naomi"https://www.zbmath.org/authors/?q=ai:feldheim.naomi-dvora"Feldheim, Ohad"https://www.zbmath.org/authors/?q=ai:feldheim.ohad-noy"Jaye, Benjamin"https://www.zbmath.org/authors/?q=ai:jaye.benjamin-j"Nazarov, Fedor"https://www.zbmath.org/authors/?q=ai:nazarov.fedor-l"Nitzan, Shahaf"https://www.zbmath.org/authors/?q=ai:nitzan.shahafSummary: Let \(f\) be a zero mean continuous stationary Gaussian process on \(\mathbb{R}\) whose spectral measure vanishes in a \(\delta \)-neighborhood of the origin. Then, the probability that \(f\) stays non-negative on an interval of length \(L\) is at most \(e^{-c\delta^2 L^2}\) with some absolute \(c>0\) and the result is sharp without additional assumptions.Generalized transforms and generalized convolution products associated with Gaussian paths on function space.https://www.zbmath.org/1456.600832021-04-16T16:22:00+00:00"Chang, Seung Jun"https://www.zbmath.org/authors/?q=ai:chang.seung-jun"Choi, Jae Gil"https://www.zbmath.org/authors/?q=ai:choi.jae-gilSummary: In this paper we define a more general convolution product (associated with Gaussian processes) of functionals on the function space \(C_{a,b}[0,T]\). The function space \(C_{a,b}[0,T]\) is induced by a generalized Brownian motion process. Thus the Gaussian processes used in this paper are non-centered processes. We then develop the fundamental relationships between the generalized Fourier-Feynman transform associated with the Gaussian process and the convolution product.Kloosterman paths of prime powers moduli. II.https://www.zbmath.org/1456.112332021-04-16T16:22:00+00:00"Ricotta, Guillaume"https://www.zbmath.org/authors/?q=ai:ricotta.guillaume"Royer, Emmanuel"https://www.zbmath.org/authors/?q=ai:royer.emmanuel"Shparlinski, Igor"https://www.zbmath.org/authors/?q=ai:shparlinski.igor-eSummary: \textit{G. Ricotta} and \textit{E. Royer} [Comment. Math. Helv. 93, No. 3, 493--532 (2018; Zbl 1448.11214)] have recently proved that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums \(S\left(a,b;p^n\right)/p^{n/2}\) converge in law in the Banach space of complex-valued continuous function on \([0,1]\) to an explicit random Fourier series as \((a,b)\) varies over \(\left(\mathbb{Z}/p^n\mathbb{Z}\right)^\times\times\left(\mathbb{Z}/p^n\mathbb{Z}\right)^\times,p\) tends to infinity among the odd prime numbers and \(n\geq 2\) is a fixed integer. This is the analogue of the result obtained by \textit{E.~Kowalski} and \textit{W. F. Sawin} [Compos. Math. 152, No. 7, 1489--1516 (2016; Zbl 1419.11134)] in the prime moduli case.
The purpose of this work is to prove a convergence law in this Banach space as only \(a\) varies over \(\left(\mathbb{Z}/p^n\mathbb{Z}\right)^\times,p\) tends to infinity among the odd prime numbers and \(n\geq 31\) is a fixed integer.A Berry-Esseén theorem for partial sums of functionals of heavy-tailed moving averages.https://www.zbmath.org/1456.600512021-04-16T16:22:00+00:00"Basse-O'Connor, Andreas"https://www.zbmath.org/authors/?q=ai:basse-oconnor.andreas"Podolskij, Mark"https://www.zbmath.org/authors/?q=ai:podolskij.mark"Thäle, Christoph"https://www.zbmath.org/authors/?q=ai:thale.christophSummary: In this paper we obtain Berry-Esseén bounds on partial sums of functionals of heavy-tailed moving averages, including the linear fractional stable noise, stable fractional ARIMA processes and stable Ornstein-Uhlenbeck processes. Our rates are obtained for the Wasserstein and Kolmogorov distances, and depend strongly on the interplay between the memory of the process, which is controlled by a parameter \(\alpha\), and its tail-index, which is controlled by a parameter \(\beta\). In fact, we obtain the classical \(1/\sqrt{n}\) rate of convergence when the tails are not too heavy and the memory is not too strong, more precisely, when \(\alpha\beta>3\) or \(\alpha\beta>4\) in the case of Wasserstein and Kolmogorov distance, respectively.
Our quantitative bounds rely on a new second-order Poincaré inequality on the Poisson space, which we derive through a combination of Stein's method and Malliavin calculus. This inequality improves and generalizes a result by \textit{G. Last} et al. [Probab. Theory Relat. Fields 165, No. 3--4, 667--723 (2016; Zbl 1347.60012)].Condensation for random variables conditioned by the value of their sum.https://www.zbmath.org/1456.601042021-04-16T16:22:00+00:00"Godrèche, Claude"https://www.zbmath.org/authors/?q=ai:godreche.claudeContinuous-time ballistic process with random resets.https://www.zbmath.org/1456.600752021-04-16T16:22:00+00:00"Villarroel, Javier"https://www.zbmath.org/authors/?q=ai:villarroel.javier"Montero, Miquel"https://www.zbmath.org/authors/?q=ai:montero.miquelReflected quadratic BSDEs driven by \(G\)-Brownian motions.https://www.zbmath.org/1456.601422021-04-16T16:22:00+00:00"Cao, Dong"https://www.zbmath.org/authors/?q=ai:cao.dong"Tang, Shanjian"https://www.zbmath.org/authors/?q=ai:tang.shanjianSummary: In this paper, the authors consider a reflected backward stochastic differential equation driven by a \(G\)-Brownian motion \((G\)-BSDE for short), with the generator growing quadratically in the second unknown. The authors obtain the existence by the penalty method, and some a priori estimates which imply the uniqueness, for solutions of the \(G\)-BSDE. Moreover, focusing their discussion at the Markovian setting, the authors give a nonlinear Feynman-Kac formula for solutions of a fully nonlinear partial differential equation.Stochastic Loewner evolution driven by Lévy processes.https://www.zbmath.org/1456.824472021-04-16T16:22:00+00:00"Rushkin, I."https://www.zbmath.org/authors/?q=ai:rushkin.i"Oikonomou, P."https://www.zbmath.org/authors/?q=ai:oikonomou.p"Kadanoff, L. P."https://www.zbmath.org/authors/?q=ai:kadanoff.leo-p"Gruzberg, I. A."https://www.zbmath.org/authors/?q=ai:gruzberg.ilya-aLarge fluctuations and fixation in evolutionary games.https://www.zbmath.org/1456.910112021-04-16T16:22:00+00:00"Assaf, Michael"https://www.zbmath.org/authors/?q=ai:assaf.michael"Mobilia, Mauro"https://www.zbmath.org/authors/?q=ai:mobilia.mauroThe mechanism of additive composition.https://www.zbmath.org/1456.682182021-04-16T16:22:00+00:00"Tian, Ran"https://www.zbmath.org/authors/?q=ai:tian.ran"Okazaki, Naoaki"https://www.zbmath.org/authors/?q=ai:okazaki.naoaki"Inui, Kentaro"https://www.zbmath.org/authors/?q=ai:inui.kentaroSummary: Additive composition
[\textit{P. W. Foltz} et al., ``The measurement of textual coherence with latent semantic analysis'', Discourse Process 15, No. 2--3, 285--307 (1998; \url{doi:10.1080/01638539809545029});
\textit{T. K. Landauer} and \textit{S. T. Dumais}, ``A solution to Plato's problem: the latent semantic analysis theory of acquisition, induction, and representation of knowledge'', Psychol. Rev. 104, No. 2, 211--240 (1997; \url{doi:10.1037/0033-295X.104.2.211});
\textit{J. Mitchell} and \textit{M. Lapata}, ``Composition in distributional models of semantics'', Cognit. Sci. 34, No. 8, 1388--1429 (2010; \url{doi:10.1111/j.1551-6709.2010.01106.x})] is a widely used method for computing meanings of phrases, which takes the average of vector representations of the constituent words. In this article, we prove an upper bound for the bias of additive composition, which is the first theoretical analysis on compositional frameworks from a machine learning point of view. The bound is written in terms of collocation strength; we prove that the more exclusively two successive words tend to occur together, the more accurate one can guarantee their additive composition as an approximation to the natural phrase vector. Our proof relies on properties of natural language data that are empirically verified, and can be theoretically derived from an assumption that the data is generated from a Hierarchical Pitman-Yor Process. The theory endorses additive composition as a reasonable operation for calculating meanings of phrases, and suggests ways to improve additive compositionality, including: transforming entries of distributional word vectors by a function that meets a specific condition, constructing a novel type of vector representations to make additive composition sensitive to word order, and utilizing singular value decomposition to train word vectors.Scalings and fractals in information geometry: Ornstein-Uhlenbeck processes.https://www.zbmath.org/1456.601282021-04-16T16:22:00+00:00"Oxley, William"https://www.zbmath.org/authors/?q=ai:oxley.william"Kim, Eun-Jin"https://www.zbmath.org/authors/?q=ai:kim.eunjinOn the density of the supremum of the solution to the linear stochastic heat equation.https://www.zbmath.org/1456.601582021-04-16T16:22:00+00:00"Dalang, Robert C."https://www.zbmath.org/authors/?q=ai:dalang.robert-c"Pu, Fei"https://www.zbmath.org/authors/?q=ai:pu.feiThe authors are interested in the existence and properties of the probability density function of the supremum
of the solutions to SPDEs (stochastic partial differential equations). This is partly motivated by the fact that the density of the supremum of the solution is related to the study of upper bounds on hitting
probabilities for these solutions. They consider the linear stochastic heat equation with zero initial condition, either Neumann or Dirichlet boundary conditions, and Brownian sheet on \([0,\infty)\times[0,1]\) as the random noise. The mild solution is defined via the Green kernel. The goal of the paper is to establish the smoothness
of the joint density of the random vector whose components are the solution and the
supremum of an increment in time of the solution over an interval (at a fixed spatial
position), and the smoothness of the density of the supremum of the solution over
a space-time rectangle that touches the \(t = 0\) axis, using a general criterion for the smoothness
of densities for locally nondegenerate random variables. Applying the Malliavin calculus, in particular, Malliavin derivatives and properties of the
divergence operator, the authors establish a Gaussian-type upper bound on these two densities
respectively, which presents a close connection with the Hölder-continuity properties
of the solution.
Reviewer: Yuliya S. Mishura (Kyïv)Directed, cylindric and radial Brownian webs.https://www.zbmath.org/1456.602062021-04-16T16:22:00+00:00"Coupier, David"https://www.zbmath.org/authors/?q=ai:coupier.david"Marckert, Jean-François"https://www.zbmath.org/authors/?q=ai:marckert.jean-francois"Tran, Viet Chi"https://www.zbmath.org/authors/?q=ai:tran.viet-chiSummary: The Brownian web (BW) is a collection of coalescing Brownian paths \((W_{(x,t)},(x,t) \in \mathbb{R} ^2)\) indexed by the plane. It appears in particular as continuous limit of various discrete models of directed forests of coalescing random walks and navigation schemes. Radial counterparts have been considered but global invariance principles are hard to establish. In this paper, we consider cylindrical forests which in some sense interpolate between the directed and radial forests: we keep the topology of the plane while still taking into account the angular component. We define in this way the cylindric Brownian web (CBW), which is locally similar to the planar BW but has several important macroscopic differences. For example, in the CBW, the coalescence time between two paths admits exponential moments and the CBW as its dual contain each a.s. a unique bi-infinite path. This pair of bi-infinite paths is distributed as a pair of reflected Brownian motions on the cylinder. Projecting the CBW on the radial plane, we obtain a radial Brownian web (RBW), i.e. a family of coalescing paths where under a natural parametrization, the angular coordinate of a trajectory is a Brownian motion. Recasting some of the discrete radial forests of the literature on the cylinder, we propose rescalings of these forests that converge to the CBW, and deduce the global convergence of the corresponding rescaled radial forests to the RBW. In particular, a modification of the radial model proposed in \textit{C.F. Coletti} and \textit{L.A. Valencia} [``Scaling limit for a family of random paths with radial behavior'', Preprint, \url{arXiv:1310.6929}] is shown to converge to the CBW.An exponential bound for Cox regression.https://www.zbmath.org/1456.622342021-04-16T16:22:00+00:00"Goldberg, Y."https://www.zbmath.org/authors/?q=ai:goldberg.yair"Kosorok, M. R."https://www.zbmath.org/authors/?q=ai:kosorok.michael-rSummary: We present an asymptotic exponential bound for the deviation of the survival function estimator of the Cox model. We show that the bound holds even when the proportional hazards assumption does not hold.On the sub-gaussianity of the \(r\)-correlograms.https://www.zbmath.org/1456.600802021-04-16T16:22:00+00:00"Giuliano, R."https://www.zbmath.org/authors/?q=ai:giuliano.rita"Cabrera, M. Ordóñez"https://www.zbmath.org/authors/?q=ai:ordonez-cabrera.manuel-h"Volodin, A."https://www.zbmath.org/authors/?q=ai:volodin.andrej-i|volodin.andrei-iLimit theorems for record indicators in threshold \(F^\alpha \)-schemes.https://www.zbmath.org/1456.600692021-04-16T16:22:00+00:00"He, P."https://www.zbmath.org/authors/?q=ai:he.puyu|he.pimo|he.pinjie|he.peiyu|he.peiren|he.pingan|he.pengfei|he.peiling|he.paul|he.puyan|he.pei|he.peijie|he.pengcai|he.peilun|he.peng|he.ping|he.peisong|he.peijun|he.peter|he.peiying|he.peixiang|he.peipei|he.pingfan|he.pu|he.peixing|he.peikun|he.pilian|he.pengzhang|he.puhan|he.penghui"Borovkov, K. A."https://www.zbmath.org/authors/?q=ai:borovkov.konstantin-aAsymmetric Lévy flights in the presence of absorbing boundaries.https://www.zbmath.org/1456.601022021-04-16T16:22:00+00:00"de Mulatier, Clélia"https://www.zbmath.org/authors/?q=ai:de-mulatier.clelia"Rosso, Alberto"https://www.zbmath.org/authors/?q=ai:rosso.alberto"Schehr, Grégory"https://www.zbmath.org/authors/?q=ai:schehr.gregoryOn the times of attaining high levels by a random walk in a random environment.https://www.zbmath.org/1456.602682021-04-16T16:22:00+00:00"Afanasyev, V. I."https://www.zbmath.org/authors/?q=ai:afanasev.valerii-ivanovichSharp weak-type \((p,p)\) estimates \((1<p<\infty)\) for positive dyadic shifts.https://www.zbmath.org/1456.420232021-04-16T16:22:00+00:00"Osękowski, Adam"https://www.zbmath.org/authors/?q=ai:osekowski.adamSummary: The paper contains the study of sharp weak-type estimates for positive dyadic shifts on \(\mathbb{R}^d\). The proof exploits Bellman function method: the inequalities are deduced from the existence of certain associated special functions, enjoying appropriate majorization and concavity conditions.Non-regular \(g\)-measures and variable length memory chains.https://www.zbmath.org/1456.600792021-04-16T16:22:00+00:00"Ferreira, Ricardo F."https://www.zbmath.org/authors/?q=ai:ferreira.ricardo-f"Gallo, Sandro"https://www.zbmath.org/authors/?q=ai:gallo.sandro"Paccaut, Frédéric"https://www.zbmath.org/authors/?q=ai:paccaut.fredericStrong convergence theorem for Walsh-Kaczmarz-Fejér means.https://www.zbmath.org/1456.420352021-04-16T16:22:00+00:00"Gogolashvili, Nata"https://www.zbmath.org/authors/?q=ai:gogolashvili.nata"Nagy, Károly"https://www.zbmath.org/authors/?q=ai:nagy.karoly"Tephnadze, George"https://www.zbmath.org/authors/?q=ai:tephnadze.georgeSummary: As main result we prove that Fejér means of Walsh-Kaczmarz-Fourier series are uniformly bounded operators from the Hardy martingale space \(H_p\) to the Hardy martingale space \(H_p\) for \(0<p\leq 1/2\).Asymptotic theory for near integrated processes driven by tempered linear processes.https://www.zbmath.org/1456.622152021-04-16T16:22:00+00:00"Sabzikar, Farzad"https://www.zbmath.org/authors/?q=ai:sabzikar.farzad"Wang, Qiying"https://www.zbmath.org/authors/?q=ai:wang.qiying"Phillips, Peter C. B."https://www.zbmath.org/authors/?q=ai:phillips.peter-c-bSummary: In an early article on near-unit root autoregression, \textit{J. Ahtola} and \textit{G. C. Tiao} [``Parameter inference for a nearly nonstationary first order autoregressive model'', Biometrika 71, No. 2, 263--272 (1984; \url{doi:10.1093/biomet/71.2.263})] studied the behavior of the score function in a stationary first order autoregression driven by independent Gaussian innovations as the autoregressive coefficient approached unity from below. The present paper develops asymptotic theory for near-integrated random processes and associated regressions including the score function in more general settings where the errors are tempered linear processes. Tempered processes are stationary time series that have a semi-long memory property in the sense that the autocovariogram of the process resembles that of a long memory model for moderate lags but eventually diminishes exponentially fast according to the presence of a decay factor governed by a tempering parameter. When the tempering parameter is sample size dependent, the resulting class of processes admits a wide range of behavior that includes both long memory, semi-long memory, and short memory processes. The paper develops asymptotic theory for such processes and associated regression statistics thereby extending earlier findings that fall within certain subclasses of processes involving near-integrated time series. The limit results relate to tempered fractional processes that include tempered fractional Brownian motion and tempered fractional diffusions of the second kind. The theory is extended to provide the limiting distribution for autoregressions with such tempered near-integrated time series, thereby enabling analysis of the limit properties of statistics of particular interest in econometrics, such as unit root tests, under more general conditions than existing theory. Some extensions of the theory to the multivariate case are reported.Superdiffusion driven by exponentially decaying memory.https://www.zbmath.org/1456.828292021-04-16T16:22:00+00:00"Alves, G. A."https://www.zbmath.org/authors/?q=ai:alves.gilvan-a"de Araújo, J. M."https://www.zbmath.org/authors/?q=ai:de-araujo.jose-milton"Cressoni, J. C."https://www.zbmath.org/authors/?q=ai:cressoni.j-c"da Silva, L. R."https://www.zbmath.org/authors/?q=ai:da-silva.luciano-r"da Silva, M. A. A."https://www.zbmath.org/authors/?q=ai:da-silva.marco-antonio-alves"Viswanathan, G. M."https://www.zbmath.org/authors/?q=ai:viswanathan.gandhimohan-mNote on a q-modified central limit theorem.https://www.zbmath.org/1456.600602021-04-16T16:22:00+00:00"Hilhorst, H. J."https://www.zbmath.org/authors/?q=ai:hilhorst.hendrik-janTesting serial correlations in high-dimensional time series via extreme value theory.https://www.zbmath.org/1456.622222021-04-16T16:22:00+00:00"Tsay, Ruey S."https://www.zbmath.org/authors/?q=ai:tsay.ruey-sSummary: This paper proposes a simple test for detecting serial correlations in high-dimensional time series. The proposed test makes use of the robust properties of Spearman's rank correlation and the theory of extreme values. Asymptotic properties of the test statistics are derived under some minor conditions as both the sample size and dimension go to infinity. The test is not sensitive to the underlying distribution of the time series so long as the data are continuously distributed. In particular, the existence of finite-order moments of the underlying distribution is not required, and asymptotic critical values of the test statistics are available in closed form. In finite samples, we correct biases of the sample autocorrelations and conduct simulations to study the performance of the proposed test statistics. Simulation results show that the proposed test statistics enjoy good properties of size and power in finite samples. We apply the proposed test to a 92-dimensional series of asset returns. Finally, a simple R code is available to obtain finite-sample critical values of the test statistics if needed.Lack-of-fit of a parametric measurement error AR(1) model.https://www.zbmath.org/1456.621782021-04-16T16:22:00+00:00"Balakrishna, N."https://www.zbmath.org/authors/?q=ai:balakrishna.naveen|balakrishna.narayana"Kim, Jiwoong"https://www.zbmath.org/authors/?q=ai:kim.jiwoong"Koul, Hira L."https://www.zbmath.org/authors/?q=ai:koul.hira-lalSummary: This paper proposes an asymptotically distribution free test for fitting a parametric model to the autoregressive function in the AR(1) model in the presence of measurement error. The test is based on a martingale transform of a certain marked residual empirical process. A simulation study assessing the finite sample level and power performance of the proposed test is also included.Records and sequences of records from random variables with a linear trend.https://www.zbmath.org/1456.601252021-04-16T16:22:00+00:00"Franke, Jasper"https://www.zbmath.org/authors/?q=ai:franke.jasper"Wergen, Gregor"https://www.zbmath.org/authors/?q=ai:wergen.gregor"Krug, Joachim"https://www.zbmath.org/authors/?q=ai:krug.joachimSome results on the Brownian meander with drift.https://www.zbmath.org/1456.600902021-04-16T16:22:00+00:00"Iafrate, F."https://www.zbmath.org/authors/?q=ai:iafrate.francesco"Orsingher, E."https://www.zbmath.org/authors/?q=ai:orsingher.enzoSummary: In this paper we study the drifted Brownian meander that is a Brownian motion starting from \(u\) and subject to the condition that \(\min_{ 0\le z\le t}B(z)> v\) with \(u > v \). The limiting process for \(u\downarrow v\) is analysed, and the sufficient conditions for its construction are given. We also study the distribution of the maximum of the meander with drift and the related first-passage times. The representation of the meander endowed with a drift is provided and extends the well-known result of the driftless case. The last part concerns the drifted excursion process the distribution of which coincides with the driftless case.Regularly varying random fields.https://www.zbmath.org/1456.601272021-04-16T16:22:00+00:00"Wu, Lifan"https://www.zbmath.org/authors/?q=ai:wu.lifan"Samorodnitsky, Gennady"https://www.zbmath.org/authors/?q=ai:samorodnitsky.gennady-pSummary: We study the extremes of multivariate regularly varying random fields. The crucial tools in our study are the tail field and the spectral field, notions that extend the tail and spectral processes of \textit{B. Basrak} and \textit{J. Segers} [Stochastic Processes Appl. 119, No. 4, 1055--1080 (2009; Zbl 1161.60319)]. The spatial context requires multiple notions of extremal index, and the tail and spectral fields are applied to clarify these notions and other aspects of extremal clusters. An important application of the techniques we develop is to the Brown-Resnick random fields.Estimates on the tail probabilities of subordinators and applications to general time fractional equations.https://www.zbmath.org/1456.601182021-04-16T16:22:00+00:00"Cho, Soobin"https://www.zbmath.org/authors/?q=ai:cho.soobin"Kim, Panki"https://www.zbmath.org/authors/?q=ai:kim.pankiSummary: In this paper, we study estimates on tail probabilities of several classes of subordinators under mild assumptions on the tails of their Lévy measures. As an application of that result, we obtain two-sided estimates for fundamental solutions of general homogeneous time fractional equations including those with Dirichlet boundary conditions.Asymptotics of intersection local time for diffusion processes.https://www.zbmath.org/1456.601952021-04-16T16:22:00+00:00"Dorogovtsev, Andrey"https://www.zbmath.org/authors/?q=ai:dorogovtsev.andrey-a"Izyumtseva, Olga"https://www.zbmath.org/authors/?q=ai:izyumtseva.olgaSummary: In the paper, we investigate the intersection local time for two correlated Brownian motions on the plane that form a diffusion process in \(\mathbb{R}^4\) associated with a divergence-form generator. Using Gaussian heat kernel bounds, we prove the existence of intersection local time for these Brownian motions, obtain estimates of its moments, and establish the law of iterated logarithm for it.Extreme value statistics from the real space renormalization group: Brownian motion, Bessel processes and continuous time random walks.https://www.zbmath.org/1456.824052021-04-16T16:22:00+00:00"Schehr, Grégory"https://www.zbmath.org/authors/?q=ai:schehr.gregory"Le Doussal, Pierre"https://www.zbmath.org/authors/?q=ai:le-doussal.pierreMinimax rates for the covariance estimation of multi-dimensional Lévy processes with high-frequency data.https://www.zbmath.org/1456.601162021-04-16T16:22:00+00:00"Papagiannouli, Katerina"https://www.zbmath.org/authors/?q=ai:papagiannouli.katerinaSummary: This article studies nonparametric methods to estimate the co-integrated volatility of multi-dimensional Lévy processes with high frequency data. We construct a spectral estimator for the co-integrated volatility and prove minimax rates for an appropriate bounded nonparametric class of Lévy processes. Given \(n\) observations of increments over intervals of length \(1/n\), the rates of convergence are \(1/\sqrt{n}\) if \(r\leq 1\) and \((n\log n)^{(r-2)/2}\) if \(r>1\), where \(r\) is the co-jump activity index and corresponds to the intensity of dependent jumps. These rates are optimal in a minimax sense. We bound the co-jump activity index from below by the harmonic mean of the jump activity indices of the components. Finally, we assess the efficiency of our estimator by comparing it with estimators in the existing literature.The Pearson walk with shrinking steps in two dimensions.https://www.zbmath.org/1456.824492021-04-16T16:22:00+00:00"Serino, C. A."https://www.zbmath.org/authors/?q=ai:serino.c-a"Redner, S."https://www.zbmath.org/authors/?q=ai:redner.sidneyCramér moderate deviation expansion for martingales with one-sided Sakhanenko's condition and its applications.https://www.zbmath.org/1456.600982021-04-16T16:22:00+00:00"Fan, Xiequan"https://www.zbmath.org/authors/?q=ai:fan.xiequan"Grama, Ion"https://www.zbmath.org/authors/?q=ai:grama.ion-g"Liu, Quansheng"https://www.zbmath.org/authors/?q=ai:liu.quanshengSummary: We give a Cramér moderate deviation expansion for martingales with differences having finite conditional moments of order \(2+\rho,\rho\in(0,1]\), and finite one-sided conditional exponential moments. The upper bound of the range of validity and the remainder of our expansion are both optimal. Consequently, our result leads to a one-sided moderate deviation principle for martingales. Moreover, applications to quantile coupling inequality, \(\beta\)-mixing sequences and \(\psi\)-mixing sequences are discussed.Law of the first passage triple of a spectrally positive strictly stable process.https://www.zbmath.org/1456.601122021-04-16T16:22:00+00:00"Chi, Zhiyi"https://www.zbmath.org/authors/?q=ai:chi.zhiyiSummary: For a spectrally positive and strictly stable process with index in (1, 2), a series representation is obtained for the joint distribution of the ``first passage triple'' that consists of the time of first passage and the undershoot and the overshoot at first passage. The result leads to several corollaries, including (1) the joint law of the first passage triple and the pre-passage running supremum, and (2) at a fixed time point, the joint law of the process' value, running supremum, and the time of the running supremum. The representation can be decomposed as a sum of strictly positive functions that allow exact sampling of the first passage triple.Clustering of extreme events in time series generated by the fractional Ornstein-Uhlenbeck equation.https://www.zbmath.org/1456.622212021-04-16T16:22:00+00:00"Telesca, Luciano"https://www.zbmath.org/authors/?q=ai:telesca.luciano"Czechowski, Zbigniew"https://www.zbmath.org/authors/?q=ai:czechowski.zbigniew.1Summary: We analyze the time clustering phenomenon in sequences of extremes of time series generated by the fractional Ornstein-Uhlenbeck (fO-U) equation as the source of long-term correlation. We used the percentile-based definition of extremes based on the crossing theory or run theory, where a \textit{run} is a sequence of \(L\) contiguous values above a given percentile. Thus, a sequence of extremes becomes a point process in time, being the time of occurrence of the extreme the starting time of the run. We investigate the relationship between the Hurst exponent related to the time series generated by the fO-U equation and three measures of time clustering of the corresponding extremes defined on the base of the 95th percentile. Our results suggest that for persistent pure fractional Gaussian noise, the sequence of the extremes is clusterized, while extremes obtained by antipersistent or Markovian pure fractional Gaussian noise seem to behave more regularly or Poissonianly. However, for the fractional Ornstein-Uhlenbeck equation, the clustering of extremes is evident even for antipersistent and Markovian cases. This is a result of short range correlations caused by differential and drift terms. The drift parameter influences the extremes clustering effect -- it drops with increasing value of the parameter.
{\copyright 2020 American Institute of Physics}Density bounds for solutions to differential equations driven by Gaussian rough paths.https://www.zbmath.org/1456.600842021-04-16T16:22:00+00:00"Gess, Benjamin"https://www.zbmath.org/authors/?q=ai:gess.benjamin"Ouyang, Cheng"https://www.zbmath.org/authors/?q=ai:ouyang.cheng"Tindel, Samy"https://www.zbmath.org/authors/?q=ai:tindel.samySummary: We consider finite-dimensional rough differential equations driven by centered Gaussian processes. Combining Malliavin calculus, rough paths techniques and interpolation inequalities, we establish upper bounds on the density of the corresponding solution for any fixed time \(t>0\). In addition, we provide Varadhan estimates for the asymptotic behavior of the density for small noise. The emphasis is on working with general Gaussian processes with covariance function satisfying suitable abstract, checkable conditions.Law of two-sided exit by a spectrally positive strictly stable process.https://www.zbmath.org/1456.601132021-04-16T16:22:00+00:00"Chi, Zhiyi"https://www.zbmath.org/authors/?q=ai:chi.zhiyiSummary: For a spectrally positive strictly stable process with index in (1,2), we obtain (i) the sub-probability density of its first exit time from an interval by hitting the interval's lower end before jumping over its upper end, and (ii) the joint distribution of the time, undershoot, and jump of the process when it makes the first exit the other way around. The density of the exit time is expressed in terms of the roots of a Mittag-Leffler function. Some theoretical applications of the results are given.Mean conservation of nodal volume and connectivity measures for Gaussian ensembles.https://www.zbmath.org/1456.824542021-04-16T16:22:00+00:00"Beliaev, Dmitry"https://www.zbmath.org/authors/?q=ai:beliaev.dmitri-b"Muirhead, Stephen"https://www.zbmath.org/authors/?q=ai:muirhead.stephen"Wigman, Igor"https://www.zbmath.org/authors/?q=ai:wigman.igorSummary: We study in depth the nesting graph and volume distribution of the nodal domains of a Gaussian field, which have been shown in previous works to exhibit asymptotic laws. A striking link is established between the asymptotic mean connectivity of a nodal domain (i.e. the vertex degree in its nesting graph) and the positivity of the percolation probability of the field, along with a direct dependence of the average nodal volume on the percolation probability. Our results support the prevailing ansatz that the mean connectivity and volume of a nodal domain is conserved for generic random fields in dimension \(d = 2\) but not in \(d \geq 3\), and are applied to a number of concrete motivating examples.Invariance principles for random walks in cones.https://www.zbmath.org/1456.601032021-04-16T16:22:00+00:00"Duraj, Jetlir"https://www.zbmath.org/authors/?q=ai:duraj.jetlir"Wachtel, Vitali"https://www.zbmath.org/authors/?q=ai:wachtel.vitali-iSummary: We prove invariance principles for a multidimensional random walk conditioned to stay in a cone. Our first result concerns convergence towards the Brownian meander in the cone. Furthermore, we prove functional convergence of \(h\)-transformed random walk to the corresponding \(h\)-transform of the Brownian motion. Finally, we prove an invariance principle for bridges of a random walk in a cone.Forward-backward SDEs with jumps and classical solutions to nonlocal quasilinear parabolic PDEs.https://www.zbmath.org/1456.601622021-04-16T16:22:00+00:00"Shamarova, Evelina"https://www.zbmath.org/authors/?q=ai:shamarova.evelina"Sá Pereira, Rui"https://www.zbmath.org/authors/?q=ai:sa-pereira.ruiSummary: We obtain an existence and uniqueness theorem for fully coupled forward-backward SDEs (FBSDEs) with jumps via the classical solution to the associated quasilinear parabolic partial integro-differential equation (PIDE), and provide the explicit form of the FBSDE solution. Moreover, we embed the associated PIDE into a suitable class of non-local quasilinear parabolic PDEs which allows us to extend the methodology of \textit{O. A. Ladyzhenskaya} et al. [Linear and quasi-linear equations of parabolic type. Translated from the Russian by S. Smith. American Mathematical Society (AMS), Providence, RI (1968; Zbl 0174.15403)] to non-local PDEs of this class. Namely, we obtain the existence and uniqueness of a classical solution to both the Cauchy problem and the initial-boundary value problem for non-local quasilinear parabolic second-order PDEs.Anisotropic scaling limits of long-range dependent random fields.https://www.zbmath.org/1456.601232021-04-16T16:22:00+00:00"Surgailis, Donatas"https://www.zbmath.org/authors/?q=ai:surgailis.donatasSummary: We review recent results on anisotropic scaling limits and the scaling transition for linear and their subordinated nonlinear long-range dependent stationary random fields \(X\) on \(\mathbb{Z}^2\). The scaling limits \({V}_{\gamma}^X\) are taken over rectangles in \(\mathbb{Z}^2\) whose sides increase as \(O( \lambda )\) and \(O(\lambda \gamma \) ) as \(\lambda \rightarrow \infty\) for any fixed \(\gamma > 0\). The scaling transition occurs at \({\gamma}_0^X>0\) provided that \({V}_{\gamma}^X\) are different for \(\gamma >{\gamma}_0^X\) and \(\gamma <{\gamma}_0^X\) and do not depend on \(\gamma\) otherwise.The probability companion for engineering and computer science.https://www.zbmath.org/1456.600012021-04-16T16:22:00+00:00"Prügel-Bennett, Adam"https://www.zbmath.org/authors/?q=ai:prugel-bennett.adamFrom the cover of the book: ``This friendly guide is the companion you need to convert pure mathematics into understanding and facility with a host of probabilistic tools. The book provides a high-level view of probability and its most powerful applications. It begins with the basic rules of probability and quickly progresses to some of the most sophisticated modern techniques in use, including Kalman filters, Monte Carlo techniques, machine learning methods, Bayesian inference and stochastic processes. It draws on thirty years of experience in applying probabilistic methods to problems in computational science and engineering, and numerous practical examples illustrate where these techniques are used in the real world. Topics of discussion range from carbon dating to Wasserstein GANs, one of the most recent developments in deep learning. The underlying mathematics is presented in full, but clarity takes priority over complete rigour, making this text a starting reference source for researchers and a readable overview for students.''
The book is very large structured in the Preface, Nomenclature, 12 chapters (divided in 64 subchapters), Appendix A (divided in 12 subchapters), Appendix B (divided in 3 subchapters), Bibliography, Index:
Chapter 1. Introduction -- Chapter 2. Survey of distributions -- Chapter 3. Monte Carlo -- Chapter 4. Discrete random variables -- Chapter 5. The normal distribution -- Chapter 6. Handling experimental data -- Chapter 7. Mathematics of random variables -- Chapter 8. Bayes -- Chapter 9. Entropy -- Chapter 10. Collective behaviour -- Chapter 11. Markov chains -- Chapter 12. Stochastic processes -- Appendix A: Answers to exercises -- Appendix B: Probability distributions.
All the chapters contain examples and finish with exercises, thus we have more than 60 problems for solving. Most of the chapters contain hints for additional reading. The bibliography contains more than 70 references and the index more than 360 items. The short evaluations of the individual references in the bibliography are worth mentioning.
New in the book is the connection to machine learning methods, cp. Subchapter 8.5: Machine learning. The author wrote on page 254: ``There are an enormous number of books on Bayesian approaches to machine learning'', e.g., in the bibliography [\textit{D. Barber}, Bayesian reasoning and machine learning. Cambridge: Cambridge University Press (2012; Zbl 1267.68001); \textit{C. M. Bishop}, Pattern recognition and machine learning. New York, NY: Springer (2006; Zbl 1107.68072); \textit{C. E. Rasmussen} and \textit{C. K. I. Williams}, Gaussian processes for machine learning. Cambridge, MA: MIT Press (2006; Zbl 1177.68165); \textit{J. Pearl}, Probabilistic reasoning in intelligent systems: networks of plausible inference. San Mateo etc.: Morgan Kaufmann Publishers (1989; Zbl 0746.68089); \textit{D. J. C. MacKay}, Information theory, inference and learning algorithms. Cambridge: Cambridge University Press (2003; Zbl 1055.94001)].
The book can be very recommended all readers, who are interested in this field.
Reviewer: Ludwig Paditz (Dresden)Positive harmonic functions on groups and covering spaces.https://www.zbmath.org/1456.530532021-04-16T16:22:00+00:00"Polymerakis, Panagiotis"https://www.zbmath.org/authors/?q=ai:polymerakis.panagiotisSummary: We show that if \(p : M \to N\) is a normal Riemannian covering, with \(N\) closed, and \(M\) has exponential volume growth, then there are non-constant, positive harmonic functions on \(M\). This was conjectured by \textit{T. Lyons} and \textit{D. Sullivan} [J. Differ. Geom. 19, 299--323 (1984; Zbl 0554.58022)].Local probabilities of randomly stopped sums of power-law lattice random variables.https://www.zbmath.org/1456.601012021-04-16T16:22:00+00:00"Bloznelis, Mindaugas"https://www.zbmath.org/authors/?q=ai:bloznelis.mindaugasSummary: Let \(X_1\) and \(N \geq 0\) be integer-valued power-law random variables. For a randomly stopped sum \(S_N = X_1+\cdots+X_N\) of independent and identically distributed copies of \(X_1\), we establish a first-order asymptotics of the local probabilities \(\mathbf{P}(S_N = t)\) as \(t \rightarrow + \infty \). Using this result, we show the scaling \(k^{- \delta }\), \(0 \leq \delta \leq 1\), of the local clustering coefficient (of a randomly selected vertex of degree \(k)\) in a power-law affiliation network.