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Heavy-tail phenomena. Probabilistic and statistical modeling. (English) Zbl 1152.62029
Springer Series in Operations Research and Financial Engineering. New York, NY: Springer (ISBN 0-387-24272-4/hbk). xix, 404 p. (2007).
Heavy tails are characteristic of many phenomena where the probability of a single huge value impacts heavily. Record-breaking insurance losses, financial-log returns, files sizes stored on a server, transmission rates of files are all examples of heavy-tail phenomena. This comprehensive text gives an interesting and useful blend of the mathematical, probabilistic and statistical tools used in heavy-tail analysis.
Chapter 1 is an introductory chapter, which gives a brief survey of some of the mathematical, probabilistic and statistical tools used in heavey-tail analysis as well as some examples of their use. Chapters 2 and 3 constitute Part I, termed crash courses. Both chapters give rapid overviews on regular variation and on weak convergence, respectively. Chapter 4 is the only chapter in Part II, termed Statistics. In this chapter, some estimators of the tail index are given, the consistency of the estimation is proved, and the effectiveness of the estimation is evaluated. The approach to inference is there semiparametric and asymptotic in nature. Some diagnostics are given that help decide on the values of the parameters and when a heavy-tail model is appropriate.
Chapters 5–8 form Part III, termed probability. Chapter 5 focuses on the Poisson process and stochastic processes derived from the Poisson process, including Lévy and extremal processes. An introduction to data network modeling is there also given. Chapter 6 discusses the dimensionless treatment of regular variation and its probabilistic equivalents. A survey on weak convergence techniques is presented and a discussion on why it is difficult to bootstrap heavy-tail phenomena is provided. Chapter 7 exploits the weak convergence technology to discuss weak convergence of extremes to extremal processes and weak convergence of summation processes to Lévy limits. Special cases include sums of heavy-tailed iid random variables converging to an alpha-stable Lévy motion. The chapter ends with a unit on how weak convergence techniques can be used to study various transformations of regularly varying random vectors. Tauberian theory for Laplace transforms is included in this discussion. Chapter 8 uses the heavy-tail machinery in service of various applied probability models of networks and queuing systems.
Chapter 9 is the only chapter in Part IV, termed more statistics, which begins with a discussion on asymptotic normality for estimators and then moves to inference for multivariate heavy-tailed models. Examples of analysis of exchange rate data, Internet data, telephone network data and insurance data are presented. The chapter ends with a discussion of the much praised and vilified sample correlation function. Part V has two appendices, devoted to notational conventions and a list of symbols, and also a section which timidly discusses some useful software. Each chapter in the Chapters 2–9 contains exercises.
Prerequisites for a reader include a prior course in stochastic processes and probability, some statistical background, some familiarity with time series analysis, and the ability to use (or at least to learn) a statistics package such as R or SPLUS. The book can serve second-year graduate students and researchers in the areas of applied mathematics, statistics, operations research, electrical engineering, and economics.

MSC:
62G32 Statistics of extreme values; tail inference
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62Pxx Applications of statistics
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
Keywords:
acf; activity rate; almost surely continuous; alternative Hill plot; angular measure; ARCH; ARMA; Arzelà-Ascoli theorem; asymptotic independence; asymptotic normality; augmentation; autocorrelation; autoregressive model; auxiliary function; binding; bootstrap; Boston University data; Breiman theorem; Brownian motion; Cauchy; centering; characteristic function; coefficient of tail dependence; compact; compactness condition; cone; subcone; connection rate; consistent estimator; continuous mapping theorem; convergence; Cramer-Wold device; crash course; Danish data; decomposition; diagnostics; differentiation; domain of attraction; Donsker’s theorem; duration; empirical measure; exceedance; exchange rate; exponent of variation; extremal process; extremes; extreme-value theory; finite-dimensional convergence; fractional Brownian motion; functional; GARCH; Glivenko-Cantelli theorem; Hausdorff metric; heavy-traffic; hidden regular variation; Hill estimator; Hill plot; homogeneous Poisson process; independence; independent increments; infinite-node Poisson model; infinite-order moving average; insurance; Internet HTTP response data; inversion; Itô construction; Karamata representation; Kolmogorov convergence criterion; Kolmogorov inequality; Laplace functional; Laplace transform; least squares; Lévy process; limit theory; limiting measure; Lindley queues; local uniform convergence; long-range dependence; marking; metric; Mittag-Leffler distribution; modulus of continuity; moment estimator; monotone function; multivariate heavy tail; multivariate regular variation; negative drift; network model; noncompliance; one-dimensional convergence; one-point uncompactification; order statistics; PP plot; partial sum; Pareto distribution; Pickands estimator; QQ estimator; QQ plot; Poisson process; Poisson transform; Portmanteau theorem; POT method; Potter bounds; PRM; Prohorov theorem; quantiles; queuing model; Radon measure; random walk; ranks; regular variation; regularly varying density; renewal theory; returns; second converging together theorem; self-similarity; semiparametric; Skorohod convergence; Slutsky’s theorem; stable Lévy motion; Stǎricǎ plot; stochastic continuity; subsequence; time-series models; topology; totally skewed; transformation; transition function; transmission rate; Urysohn lemma; vague metric; vague topology; value-at-risk; Vervaat’s lemma; von Mises condition; waiting time; weak convergence
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