Recent zbMATH articles in MSC 60B https://www.zbmath.org/atom/cc/60B 2021-11-25T18:46:10.358925Z Werkzeug Clone-induced approximation algebras of Bernoulli distributions https://www.zbmath.org/1472.08005 2021-11-25T18:46:10.358925Z "Yashunsky, Alexey D." https://www.zbmath.org/authors/?q=ai:yashunsky.aleksey-d Summary: We consider the problem of approximating distributions of Bernoulli random variables by applying Boolean functions to independent random variables with distributions from a given set. For a set $$B$$ of Boolean functions, the set of approximable distributions forms an algebra, named the approximation algebra of Bernoulli distributions induced by $$B$$. We provide a complete description of approximation algebras induced by most clones of Boolean functions. For remaining clones, we prove a criterion for approximation algebras and a property of algebras that are finitely generated. Singularity of random symmetric matrices -- simple proof https://www.zbmath.org/1472.15043 2021-11-25T18:46:10.358925Z "Ferber, Asaf" https://www.zbmath.org/authors/?q=ai:ferber.asaf Summary: In this paper we give a simple, short, and self-contained proof for a non-trivial upper bound on the probability that a random $$\pm 1$$ symmetric matrix is singular. On a theorem of Avez https://www.zbmath.org/1472.20084 2021-11-25T18:46:10.358925Z "Elder, Murray" https://www.zbmath.org/authors/?q=ai:elder.murray-j "Rogers, Cameron" https://www.zbmath.org/authors/?q=ai:rogers.cameron Summary: For each symmetric, aperiodic probability measure $$\mu$$ on a finitely generated group $$G$$, we define a subset $$A_{\mu}$$ consisting of group elements $$g$$ for which the limit of the ratio $$\mu^{\ast n}(g)/\mu^{\ast n}(e)$$ tends to 1. We prove that $$A_{\mu}$$ is a subgroup, is amenable, contains every finite normal subgroup, and $$G=A_{\mu}$$ if and only if $$G$$ is amenable. For non-amenable groups we show that $$A_{\mu}$$ is not always a normal subgroup and can depend on the measure. We formulate some conjectures relating $$A_{\mu}$$ to the amenable radical. Positive harmonic functions on the Heisenberg group. II https://www.zbmath.org/1472.31012 2021-11-25T18:46:10.358925Z "Benoist, Yves" https://www.zbmath.org/authors/?q=ai:benoist.yves This article discusses extremal positive harmonic functions for finitely supported measures on the discrete Heisenberg group. The author establishes that extremal positive harmonic functions are proportional either to characters or to translates of induced from characters. For Part I, see [the author, Positive Harmonic Functions on the Heisenberg group. I'', Preprint, \url{arXiv:1907.05041}]. On the support of the free additive convolution https://www.zbmath.org/1472.46069 2021-11-25T18:46:10.358925Z "Bao, Zhigang" https://www.zbmath.org/authors/?q=ai:bao.zhigang "Erdős, László" https://www.zbmath.org/authors/?q=ai:erdos.laszlo "Schnelli, Kevin" https://www.zbmath.org/authors/?q=ai:schnelli.kevin Summary: We consider the free additive convolution of two probability measures $$\mu$$ and $$\nu$$ on the real line and show that $$\mu\boxplus v$$ is supported on a single interval if $$\mu$$ and $$\nu$$ each has single interval support. Moreover, the density of $$\mu\boxplus\nu$$ is proven to vanish as a square root near the edges of its support if both $$\mu$$ and $$\nu$$ have power law behavior with exponents between $$-1$$ and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [\textit{Z.-G. Bao} et al., J. Funct. Anal. 279, No. 7, Article ID 108639, 93~p. (2020; Zbl 1460.46058)]. An invitation to optimal transport, Wasserstein distances, and gradient flows https://www.zbmath.org/1472.49001 2021-11-25T18:46:10.358925Z "Figalli, Alessio" https://www.zbmath.org/authors/?q=ai:figalli.alessio "Glaudo, Federico" https://www.zbmath.org/authors/?q=ai:glaudo.federico This graduate text offers a relatively self-contained introduction to the optimal transport theory. It consists of five chapters and two appendices. Chapter 1 gives a brief review of the optimal transport theory, recalls certain of basics of measure theory and Riemannian geometry, and shows three typical examples of the transport maps in connection to the classical isoperimetry. Chapter 2 presents the so-called core of the optimal transport theory: the solution to Kantorovich's problem for general costs; the duality theory; the solution to Monge's problem for suitable costs. Chapter 3 utilizes the $$[1,\infty)\ni p$$-Wasserstein distances to handle an essential relationship among the optimal transport theory, gradient flows in the Hilbert spaces, and partial differential equations. Chapter 4 shows a differential viewpoint of the optimal transport theory via studying Benamou-Brenier's and Otto's formulas based on the probability measures. Chapter 5 suggests several applied topics of the optimal transport theory. Appendix A includes a set of some interesting exercises and their solutions. Appendix B outlines a proof of the disintegration theorem. Viscosity solutions for controlled McKean-Vlasov jump-diffusions https://www.zbmath.org/1472.49054 2021-11-25T18:46:10.358925Z "Burzoni, Matteo" https://www.zbmath.org/authors/?q=ai:burzoni.matteo "Ignazio, Vincenzo" https://www.zbmath.org/authors/?q=ai:ignazio.vincenzo "Reppen, A. Max" https://www.zbmath.org/authors/?q=ai:reppen.a-max "Soner, H. M." https://www.zbmath.org/authors/?q=ai:soner.halil-mete The paper deals with a class of nonlinear integro-differential equations on a subspace of all probability measures on the real line related to the optimal control of McKean-Vlasov jump-diffusions. The authors investigated an intrinsic notion of viscosity solutions that does not rely on the lifting to a Hilbert space and proved a comparison theorem for these solutions. Probability-2. Translated from the fourth Russian edition by R. P. Boas and D. M. Chibisov https://www.zbmath.org/1472.60001 2021-11-25T18:46:10.358925Z "Shiryaev, Albert N." https://www.zbmath.org/authors/?q=ai:shiryaev.albert-n This textbook is the second volume of a pair that presents the latest English edition of the author's classic, Probability. Building on the foundations established in the preceding Probability-1, this volume guides the reader on to the theory of random processes. The new edition includes expanded material on financial mathematics and financial engineering; new problems, exercises, and proofs throughout, and a historical review charting the development of the mathematical theory of probability. Suitable for an advanced undergraduate or beginning graduate student with a course in probability theory, this volume forms the natural sequel to Probability-1 [\textit{A. N. Shiryaev}, Probability-1. Translated from the fourth Russian edition by R. P. Boas and D. M. Chibisov. 3rd edition. New York, NY: Springer (2016; Zbl 1390.60002)]. Probability-2 opens with classical results related to sequences and sums of independent random variables, such as the zero-one laws, convergence of series, strong law of large numbers, and the law of the iterated logarithm. The subsequent chapters go on to develop the theory of random processes with discrete time: stationary processes, martingales, and Markov processes. The historical review illustrates the growth from intuitive notions of randomness in history through to modern day probability theory and theory of random processes. Along with its companion volume, this textbook presents a systematic treatment of probability from the ground up, starting with intuitive ideas and gradually developing more sophisticated subjects, such as random walks, martingales, Markov chains, the measure-theoretic foundations of probability theory, weak convergence of probability measures, and the central limit theorem. Many examples are discussed in detail, and there are a large number of exercises throughout. Some new material has also been added to Chapter VII that treats the theory of martingales with discrete time. In Section 9 of that chapter a discrete version of Itô's formula is presented, which may be viewed as an introduction to the stochastic calculus for the Brownian motion, where Itô's (change-of-variables) formula is of key importance. In Section 10, it is shown how the methods of the martingale theory provide a simple way of obtaining estimates of ruin probabilities for an insurance company acting under the Cramer-Lundberg model. The next Section 11 deals with the arbitrage theory'' in stochastic financial mathematics. Here he states two Fundamental theorems of the arbitrage theory'', which provide conditions in martingale terms for absence of arbitrage possibilities and conditions guaranteeing the existence of a portfolio of assets, which enables one to achieve the objected aim. Finally, Section 13 of that chapter is devoted to the general theory of optimal stopping rules for arbitrary random sequences. The material presented here demonstrates how the concepts and results of the martingale theory can be applied in the various problems of Stochastic Optimization''. There is also a number of changes and supplements made in other chapters. There is some new material concerning the theorems on monotone classes (Section 2 of Chapter II), which relies on detailed treatment of the concepts and properties of $\pi$-$\lambda$'' systems, and the fundamental theorems of mathematical statistics given in Section 13 of Chapter III. The novelty of the present edition is also the Outline of historical development of the mathematical probability theory'', placed at the end of Probability-2''. In a number of sections new problems have been added. The circular law for sparse non-Hermitian matrices https://www.zbmath.org/1472.60006 2021-11-25T18:46:10.358925Z "Basak, Anirban" https://www.zbmath.org/authors/?q=ai:basak.anirban "Rudelson, Mark" https://www.zbmath.org/authors/?q=ai:rudelson.mark Let $$\lambda_1$$,\dots, $$\lambda_n$$ the eigenvalues of a $$n\times n$$ matrix $$B$$; its empirical spectral distribution (ESD) is defined by $$L_B:=\frac{1}{n}\,\sum_{i=1}^n \delta_{\lambda_i}$$, where $$\delta_x$$ is the Dirac measure concentrated at $$x$$. The sub-Gausssian norm of a random variable $$\xi$$ is defined by $\|\xi\|_{\psi_2}:=\sup_{k\ge 1} k^{-1/2}\,\mathbb{E}^{1/k}(|\xi|^k)\,.$ The main result of this important paper is the following theorem, which extends previous results quoted in the introduction. Theorem. Let $$A_n$$ be an $$n\times n$$ matrix with i.i.d. entries $$a_{i,j}=\delta_{i,j}\,\xi_{i,j}$$, where the $$\delta_{i,j}$$ are independent Bernoulli random variables taking the value $$1$$ with probability $$p_n\in\left]0,1\right]$$ and $$\xi_{i,j}$$ are real-valued i.i.d. sub-Gaussian centred random variables with unit variance. \begin{enumerate} \item[(i)] If $$p_n$$ is such that $$np_n=\omega(\log^2 n)$$, then as $$n\to\infty$$ the \textnormal{ESD} of $$A_n/\sqrt{n\,p_n}$$ converges weakly in probability to the circular law. \item[(ii)] There exists a constant $$c$$, which depends only on the sub-Gaussian norm of $$\{\xi_{i,j}\}$$, such that if $$p_n$$ satisfies the inequality $$np_n>\exp(c\,\sqrt{\log n})$$, then the conclusion of (i) holds almost surely. \end{enumerate} Here, if $$(a_n)$$ and $$(b_n)$$ are two sequences of positive reals, one writes $$a_n=\omega(b_n)$$ if $$b_n=o(a_n)$$, $$a_n=O(b_n)$$ and $$\limsup_{n\to\infty} a_n/b_n<\infty$$ Remarks connected with the weak limit of iterates of some random-valued functions and iterative functional equations https://www.zbmath.org/1472.60007 2021-11-25T18:46:10.358925Z "Baron, Karol" https://www.zbmath.org/authors/?q=ai:baron.karol Summary: The paper consists of two parts. At first, assuming that $$(\Omega, \mathcal{A}, P)$$ is a probability space and $$(X, \varrho)$$ is a complete and separable metric space with the $$\sigma$$-algebra $$\mathcal{B}$$ of all its Borel subsets we consider the set $$\mathcal{R}_c$$ of all $$\mathcal{B} \otimes \mathcal{A}$$-measurable and contractive in mean functions $$f: X \times \Omega \rightarrow X$$ with finite integral $$\int_\Omega \varrho (f(x, \omega), x) P (d \omega)$$ for $$x \in X$$, the weak limit $$\pi^f$$ of the sequence of \textit{iterates} of $$f \in \mathcal{R}_c$$, and investigate continuity-like property of the function $$f \mapsto \pi^f$$, $$f \in \mathcal{R}_c$$, and Lipschitz solutions $$\varphi$$ that take values in a separable Banach space of the equation: $\varphi (x) = \int_\Omega \varphi (f(x,\omega) P ( d\omega) + F(x).$ Next, assuming that $$X$$ is a real separable Hilbert space, $$\Lambda$$: $$X \rightarrow X$$ is linear and continuous with $$\Vert \Lambda \Vert < 1$$, and $$\mu$$ is a probability Borel measure on $$X$$ with finite first moment we examine continuous at zero solutions $$\varphi : X \rightarrow \mathbb{C}$$ of the equation $\varphi(x) = \hat{\mu}(x)\varphi (\Lambda x)$ which characterizes the limit distribution $$\pi^{f}$$ for some special $$f \in \mathcal{R}_c$$. Traces of powers of matrices over finite fields https://www.zbmath.org/1472.60008 2021-11-25T18:46:10.358925Z "Gorodetsky, Ofir" https://www.zbmath.org/authors/?q=ai:gorodetsky.ofir "Rodgers, Brad" https://www.zbmath.org/authors/?q=ai:rodgers.brad The authors consider a prime power $$q=p^r,$$ a matrix $$M$$ chosen uniformly from the finite unitary group $$\mathrm{U}(n,q)\subset \mathrm{GL}(n,q^2),$$ and the sequence $$\{M^i\}_{1\leq i \leq k}$$ where $$i$$ is not multiple of $$p.$$ They prove that the traces of powers of matrices converge to independent uniform variables in $$\mathbb F_{q^2}$$ as $$n \rightarrow \infty.$$ The rate of convergence is shown to be exponential in $$n^2.$$ \newline The related problem of the rate at which characteristic polynomial of $$M$$ equidistributes in short intervals' of $$\mathbb F_{q^2} [T]$$ is also considered. \newline Analogous results are also proven for the general linear, special linear, symplectic and orthogonal groups over a finite field. \newline The proofs depend upon applying techniques from analytic number theory over function fields to formulas due to Fulman and others for the probability that the characteristic polynomial of a random matrix equals a given polynomial. Local laws for non-Hermitian random matrices and their products https://www.zbmath.org/1472.60009 2021-11-25T18:46:10.358925Z "Götze, Friedrich" https://www.zbmath.org/authors/?q=ai:gotze.friedrich-w "Naumov, Alexey" https://www.zbmath.org/authors/?q=ai:naumov.a-a "Tikhomirov, Alexander" https://www.zbmath.org/authors/?q=ai:tikhomirov.alexander-n Partial generalized four moment theorem revisited https://www.zbmath.org/1472.60010 2021-11-25T18:46:10.358925Z "Jiang, Dandan" https://www.zbmath.org/authors/?q=ai:jiang.dandan "Bai, Zhidong" https://www.zbmath.org/authors/?q=ai:bai.zhi-dong Summary: This is a complementary proof of partial generalized 4 moment theorem (PG4MT) mentioned and described by the authors [ibid. 27, No. 1, 274--294 (2021; Zbl 07282851)]. Since the G4MT proposed in that paper requires both the matrices $$\mathbf{X}$$ and $$\mathbf{Y}$$ satisfying the assumption $$\max_{t,s}|u_{ts}|^2\mathrm{E}\{|x_{11}|^4I(|x_{11}|<\sqrt{n})-\mu\}\to 0$$ with the same $$\mu$$ which maybe restrictive in real applications, we proposed a new G4MT, called PG4MT, without proof. After the manuscript posed in ArXiv, the authors received high interests in the proof of PG4MT through private communications and find the PG4MT more general than G4MT, it is necessary to give a detailed proof of it. Moreover, it is found that the PG4MT derives a CLT of spiked eigenvalues of sample covariance matrices which covers the work in [the first author and \textit{J. Yao}, J. Multivariate Anal. 106, 167--177 (2012; Zbl 1301.62049)] as a special case. The smallest eigenvalue distribution of the Jacobi unitary ensembles https://www.zbmath.org/1472.60011 2021-11-25T18:46:10.358925Z "Lyu, Shulin" https://www.zbmath.org/authors/?q=ai:lyu.shulin "Chen, Yang" https://www.zbmath.org/authors/?q=ai:chen.yang.1 Summary: In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight $$x^{\alpha}(1 - x)^{\beta}, x \in [0, 1], \alpha, \beta > -1$$, the probability that all eigenvalues of Hermitian matrices from this ensemble lie in the interval $$[t, 1]$$ is given by the Fredholm determinant of the Bessel kernel. We derive the constant in the asymptotics of this Bessel kernel determinant. A specialization of the results gives the constant in the asymptotics of the probability that the interval $$(- a, a), a > 0$$ is free of eigenvalues in the Jacobi unitary ensemble with the symmetric weight $$(1 - x^2)^{\beta}, x \in [- 1, 1]$$. Large deviations for extreme eigenvalues of deformed Wigner random matrices https://www.zbmath.org/1472.60012 2021-11-25T18:46:10.358925Z "Mckenna, Benjamin" https://www.zbmath.org/authors/?q=ai:mckenna.benjamin The purpose of the paper is to prove a large deviation principle (LDP) for the largest eigenvalue of the random matrix $${X_N} = \frac{{{W_N}}}{{\sqrt N }} + {D_N}$$, where $$\frac{{{W_N}}}{{\sqrt N }}$$ lies in a particular class of real or complex Wigner matrices. In particular this includes Gaussian ensembles with full-rank general deformation. For the non-Gaussian ensembles, the deformation should be diagonal, and the laws of the entries of $${W_N}$$ are supposed to have sharp sub-Gaussian Laplace transforms and satisfy certain concentration properties. It is also assumed that $${D_N}$$ is a deterministic matrix whose empirical spectral measure tends to a deterministic limit $${\mu _D}$$ and whose extreme eigenvalues tend to the edges of $${\mu _D}$$. For these ensembles the paper establishes LDP in a restricted range $$( - \infty ,{x_c})$$, where $${x_c}$$ depends on the deformation only and can be infinite. Concentration inequalities for random tensors https://www.zbmath.org/1472.60041 2021-11-25T18:46:10.358925Z "Vershynin, Roman" https://www.zbmath.org/authors/?q=ai:vershynin.roman Let $$x_1,x_2,\ldots$$ be independent random vectors in $$\mathbb{R}^n$$ whose coordinates are independent random variables with zero mean and unit variance, and let $$X=x_1\otimes\cdots\otimes x_d$$, a random tensor in $$\mathbb{R}^{n^d}$$. The author proves two concentration inequalities for $$X$$. Firstly, in the case where the $$x_k$$ are bounded almost surely, it is shown that for a convex and Lipschitz function $$f$$, and all $$0\leq t\leq2(\mathbb{E}|f(X)|^2)^{1/2}$$, we have $\mathbb{P}\left(\big|f(X)-\mathbb{E}f(X)\big|>t\right)\leq2\exp\left(-\frac{ct^2}{dn^{d-1}\|f\|^2_{Lip}}\right)\,,$ for some constant $$c>0$$ depending on the bound for the $$x_k$$. Secondly, in the case where the $$x_k$$ are sub-Gaussian, it is shown that for a linear operator $$A$$ taking values in a Hilbert space $$H$$, and all $$0\leq t\leq2\|A\|_{HS}$$, we have $\mathbb{P}\left(\big|\|AX\|_H-\|A\|_{HS}\big|\geq t\right)\leq2\exp\left(-\frac{ct^2}{dn^{d-1}\|A\|^2_{op}}\right)\,,$ where $$c>0$$ again depends on the $$x_k$$, and where $$\|A\|_{HS}$$ and $$\|A\|_{op}$$ are the Hilbert-Schmidt and operator norms of $$A$$, respectively. As an application of this latter concentration bound, the author shows that random tensors are well conditioned; that is, if $$d=o(\sqrt{n/\log(n)})$$ then with high probability $$(1-o(1))n^d$$ independent copies of $$X$$ are far from linearly dependent. Contraction principle for trajectories of random walks and Cramér's theorem for kernel-weighted sums https://www.zbmath.org/1472.60053 2021-11-25T18:46:10.358925Z "Vysotsky, Vladislav" https://www.zbmath.org/authors/?q=ai:vysotsky.vladislav-v Summary: In 2013 \textit{A. A. Borovkov} and \textit{A. A. Mogulskii} [Theory Probab. Appl. 57, No. 1, 1--27 (2013; Zbl 1279.60037); translation from Teor. Veroyatn. Primen. 57, No. 1, 3--34 (2012)] proved a weaker-than-standard `metric'' large deviations principle (LDP) for trajectories of random walks in $$\mathbb{R}^d$$ whose increments have the Laplace transform finite in a neighbourhood of zero. We prove that general metric LDPs are preserved under uniformly continuous mappings. This allows us to transform the result of Borovkov and Mogulskii into standard LDPs. We also give an explicit integral representation of the rate function they found. As an application, we extend the classical Cramér theorem by proving an LPD for kernel-weighted sums of i.i.d. random vectors in $$\mathbb{R}^d$$. Equidistribution of random walks on compact groups. II: The Wasserstein metric https://www.zbmath.org/1472.60077 2021-11-25T18:46:10.358925Z "Borda, Bence" https://www.zbmath.org/authors/?q=ai:borda.bence Summary: We consider a random walk $$S_k$$ with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum $$\sum_{k=1}^Nf(S_k)$$ with Hölder continuous test functions $$f$$, including the central limit theorem, the law of the iterated logarithm and an almost sure approximation by a Wiener process, provided that the distribution of $$S_k$$ converges to the Haar measure in the $$p$$-Wasserstein metric fast enough. As an example, we construct discrete random walks on an irrational lattice on the torus $$\mathbb{R}^d/\mathbb{Z}^d$$, and find their precise rate of convergence to uniformity in the $$p$$-Wasserstein metric. The proof uses a new Berry-Esseen type inequality for the $$p$$-Wasserstein metric on the torus, and the simultaneous Diophantine approximation properties of the lattice. These results complement the first part of this paper on random walks with an absolutely continuous component and quantitative ergodic theorems for Borel measurable test functions. For Part I, see [the author, Ann. Inst. Henri Poincaré, Probab. Stat. 57, No. 1, 54--72 (2021; Zbl 1468.60054)]. Cutoff for product replacement on finite groups https://www.zbmath.org/1472.60120 2021-11-25T18:46:10.358925Z "Peres, Yuval" https://www.zbmath.org/authors/?q=ai:peres.yuval "Tanaka, Ryokichi" https://www.zbmath.org/authors/?q=ai:tanaka.ryokichi "Zhai, Alex" https://www.zbmath.org/authors/?q=ai:zhai.alex Let $$G$$ be a finite group, $$[n]:=\{1,2,\cdots,n\}$$, and $$G^n$$ be the set of all functions $$\sigma: [n]\to G$$. Denote by $$\mathcal{S}$$ the space of generating $$n$$-tuples, i.e., the set of $$\sigma$$ whose values generate $$G$$ as a group: $\mathcal{S}:=\{\sigma\in G^n: \langle \sigma(1),\dots,\sigma(n)\rangle=G\}.$ Define the so-called product replacement chain $$(\sigma_t)_{t\geq 0}$$ on $$\mathcal{S}$$ as follows: if we have a current state $$\sigma$$, then uniformly at random, choose an ordered pair $$(i,j)$$ of distinct integers in $$[n]$$, and change the value of $$\sigma(i)$$ to $$\sigma(i)\sigma(j)^{\pm 1}$$, where the signs are chosen with equal probability. This paper shows that the total-variation mixing time of the chain has a cutoff at time $$\frac{3}{2}n\log n$$ with window of order $$n$$ as $$n\to \infty$$. This extends a result of \textit{A. Ben-Hamou} and the first author [Electron. Commun. Probab. 23, Paper No. 32, 10 p. (2018; Zbl 1397.60096)] (who proved the result for $$G=\mathbb{Z}/2$$) and confirms a conjecture of \textit{P. Diaconis} and \textit{L. Saloff-Coste} [Invent. Math. 134, No. 2, 251--299 (1998; Zbl 0921.60003)] that for an arbitrary but fixed finite group, the mixing time of the product replacement chain is $$O(n\log n)$$. Unified theory for finite Markov chains https://www.zbmath.org/1472.60121 2021-11-25T18:46:10.358925Z "Rhodes, John" https://www.zbmath.org/authors/?q=ai:rhodes.john-l "Schilling, Anne" https://www.zbmath.org/authors/?q=ai:schilling.anne Summary: We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup $$S$$. Our methods use geometric finite semigroup theory via the Karnofsky-Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to $$P(n)$$, the set of all subsets of an $$n$$ element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or $$\mathcal{R}$$-trivial bands) by Brown and Diaconis, extensions to $$\mathcal{R}$$-trivial semigroups by Ayyer, Steinberg, Thiéry and the second author, and important recent work by Chung and Graham. The Karnofsky-Rhodes expansion of the right Cayley graph of $$S$$ in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded $$S$$. Using our previous results with Silva based on work by Berstel, Perrin, Reutenauer, we construct (infinite) semaphore codes on which we can define Markov chains. These semaphore codes can be lumped using geometric semigroup theory. Using normal forms and associated Kleene expressions, they yield formulas for the stationary distribution of the finite Markov chain of the expanded $$S$$ and the original $$S$$. Analyzing the normal forms also provides an estimate on the mixing time. Normal distributions of finite Markov chains https://www.zbmath.org/1472.60122 2021-11-25T18:46:10.358925Z "Rhodes, John" https://www.zbmath.org/authors/?q=ai:rhodes.john-l "Schilling, Anne" https://www.zbmath.org/authors/?q=ai:schilling.anne Stochastic approximation of quasi-stationary distributions for diffusion processes in a bounded domain https://www.zbmath.org/1472.60136 2021-11-25T18:46:10.358925Z "Benaïm, Michel" https://www.zbmath.org/authors/?q=ai:benaim.michel "Champagnat, Nicolas" https://www.zbmath.org/authors/?q=ai:champagnat.nicolas "Villemonais, Denis" https://www.zbmath.org/authors/?q=ai:villemonais.denis The paper studies a random process with reinforcement, which evolves following the dynamics of a given diffusion process in a bounded domain and is resampled according to its occupation measure when it reaches the boundary. They show that its occupation measure converges to the unique quasi-stationary distribution of the diffusion process absorbed at the boundary of the domain. Section 2 formulates the main assumptions and results. Section 3 contains useful general results on quasi-stationary distributions and proofs of new general results on the Green operator $$A$$ which has its own interest. Section 4 is devoted to the proof the main result, which consists in checking that the occupation measure of the resampling points is (up to a time change and linearization) an asymptotic pseudo-trajectory of a measure-valued dynamical system related to the operator $$A$$. Diffusions interacting through a random matrix: universality via stochastic Taylor expansion https://www.zbmath.org/1472.60137 2021-11-25T18:46:10.358925Z "Dembo, Amir" https://www.zbmath.org/authors/?q=ai:dembo.amir "Gheissari, Reza" https://www.zbmath.org/authors/?q=ai:gheissari.reza Summary: Consider $$(X_i(t))$$ solving a system of $$N$$ stochastic differential equations interacting through a random matrix $${\mathbf{J}} = (J_{ij})$$ with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $$(X_i(t))$$, initialized from some $$\mu$$ independent of $${\mathbf{J}}$$, are universal, i.e., only depend on the choice of the distribution $$\mathbf{J}$$ through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks. High-dimensional linear models: a random matrix perspective https://www.zbmath.org/1472.62080 2021-11-25T18:46:10.358925Z "Namdari, Jamshid" https://www.zbmath.org/authors/?q=ai:namdari.jamshid "Paul, Debashis" https://www.zbmath.org/authors/?q=ai:paul.debashis "Wang, Lili" https://www.zbmath.org/authors/?q=ai:wang.lili.3 Summary: Professor \textit{C. R. Rao}'s [Linear statistical inference and its applications. New York-London-Sydney: John Wiley and Sons, Inc. (1965; Zbl 0137.36203)] is a classic that has motivated several generations of statisticians in their pursuit of theoretical research. This paper looks into some of the fundamental problems associated with linear models, but in a scenario where the dimensionality of the observations is comparable to the sample size. This perspective, largely driven by contemporary advancements in random matrix theory, brings new insights and results that can be helpful even for solving relatively low-dimensional problems. This overview also brings into focus the fundamental roles played by the eigenvalues of large covariance-type matrices in the theory of high-dimensional multivariate statistics. Estimating leverage scores via rank revealing methods and randomization https://www.zbmath.org/1472.62082 2021-11-25T18:46:10.358925Z "Sobczyk, Aleksandros" https://www.zbmath.org/authors/?q=ai:sobczyk.aleksandros "Gallopoulos, Efstratios" https://www.zbmath.org/authors/?q=ai:gallopoulos.efstratios Covariant CP-instruments and their convolution semigroups https://www.zbmath.org/1472.81015 2021-11-25T18:46:10.358925Z "Heo, Jaeseong" https://www.zbmath.org/authors/?q=ai:heo.jaeseong "Ji, Un Cig" https://www.zbmath.org/authors/?q=ai:ji.un-cig Summary: Using probability operators and Fourier transforms of CP-instruments on von Neumann algebras, we give necessary and sufficient conditions for operators to be probability operators associated with covariant CP-instruments or to be Fourier transforms of covariant CP-instruments. We discuss a convolution semigroup of covariant CP-instruments and a semigroup of probability operators associated with CP-instruments on von Neumann algebras. Limit law of a second class particle in TASEP with non-random initial condition https://www.zbmath.org/1472.82023 2021-11-25T18:46:10.358925Z "Ferrari, P. L." https://www.zbmath.org/authors/?q=ai:ferrari.patrik-lino "Ghosal, P." https://www.zbmath.org/authors/?q=ai:ghosal.promit|ghosal.pratik|ghosal.purnata "Nejjar, P." https://www.zbmath.org/authors/?q=ai:nejjar.peter In this paper a totally asymmetric simple exclusion process (TASEP) with non-random initial condition and density $$\lambda$$ on $$\mathbb{Z}_-$$ and $$\rho$$ on $$\mathbb{Z}_+$$ as one of the simplest non-reversible interacting particle systems on $$\mathbb{Z}$$ lattice is considered. An initial and further particle configurations are assumed and described by the occupation variables $$\{\eta_j\}$$. Particles (first-class particles) can jump (they are independent) one step to the right only if their right neighboring site is empty. The particles cannot overtake each other and a labeling to them is associated. The position of particle $$k$$ at time $$t$$ is denoted by $$x_k(t)$$ with the right-to-left ordering. In this paper the second-class particles are considered: when a first-class particle tries to jump on a site occupied by a second-class particle, the jump is not suppressed and the two particles interchanges their positions. The applications of second-class particles are very often when the interacting system generates shocks as the discontinuities in the particle density. The main result of paper is given by Theorem 1.1 which is in the form of the limiting distribution and uses two ingredients: 1) the asymptotic independence of the last passage times from two disjoint initial set of points of a last passage percolation (LPP) model; 2) a tightness-type result on the two LPP problems (by Proposition 3.2 and Corollary 3.4) that extends to general the densities of the Pimentel method. The paper is divided into two sections where Section 2 shows the connection between TASEP and LPP and the proof of Theorem 1.1, which is mainly based on preliminary results on the control of LPP at different points. Volumes and random matrices https://www.zbmath.org/1472.83017 2021-11-25T18:46:10.358925Z "Witten, Edward" https://www.zbmath.org/authors/?q=ai:witten.edward Summary: This article is an introduction to newly discovered relations between volumes of moduli spaces of Riemann surfaces or super Riemann surfaces, simple models of gravity or supergravity in two dimensions, and random matrix ensembles. (The article is based on a lecture at the conference on the Mathematics of Gauge Theory and String Theory, University of Auckland, January 2020)