Stochastic Models with Power-Law Tails: The Equation X = AX + B by Dariusz Burac

Description: Stochastic Models with Power-Law Tails by Dariusz Buraczewski, Ewa Damek, Thomas Mikosch In this monograph the authors give a systematic approach to the probabilistic properties of the fixed point equation X=AX+B. A probabilistic study of the stochastic recurrence equation X_t=A_tX_{t-1}+B_t for real- and matrix-valued random variables A_t, where (A_t,B_t) constitute an iid sequence, is provided. The classical theory for these equations, including the existence and uniqueness of a stationary solution, the tail behavior with special emphasis on power law behavior, moments and support, is presented. The authors collect recent asymptotic results on extremes, point processes, partial sums (central limit theory with special emphasis on infinite variance stable limit theory), large deviations, in the univariate and multivariate cases, and they further touch on the related topics of smoothing transforms, regularly varying sequences and random iterative systems.The text gives an introduction to the Kesten-Goldie theory for stochastic recurrence equations of the type X_t=A_tX_{t-1}+B_t. It provides the classical results of Kesten, Goldie, Guivarch, and others, and gives an overview of recent results on the topic. It presents the state-of-the-art results in the field of affine stochastic recurrence equations and shows relations with non-affine recursions and multivariate regular variation. FORMAT Paperback LANGUAGE English CONDITION Brand New Back Cover In this monograph the authors give a systematic approach to the probabilistic properties of the fixed point equation X=AX+B. A probabilistic study of the stochastic recurrence equation X_t=A_tX_{t-1}+B_t for real- and matrix-valued random variables A_t, where (A_t,B_t) constitute an iid sequence, is provided. The classical theory for these equations, including the existence and uniqueness of a stationary solution, the tail behavior with special emphasis on power law behavior, moments and support, is presented. The authors collect recent asymptotic results on extremes, point processes, partial sums (central limit theory with special emphasis on infinite variance stable limit theory), large deviations, in the univariate and multivariate cases, and they further touch on the related topics of smoothing transforms, regularly varying sequences and random iterative systems. The text gives an introduction to the Kesten-Goldie theory for stochastic recurrence equations of the type X_t=A_tX_{t-1}+B_t. It provides the classical results of Kesten, Goldie, Guivarch, and others, and gives an overview of recent results on the topic. It presents the state-of-the-art results in the field of affine stochastic recurrence equations and shows relations with non-affine recursions and multivariate regular variation. Table of Contents Introduction.- The Univariate Case.- Univariate Limit Theoru.- Multivariate Case.- Miscellanea.- Appendices. Review "The authors collected together almost all the results on the stochastic recurrence equation, and on its stationary solution. … in the course of the reading we learn about Markov chains, renewal and implicit renewal theory, regular variation … point process techniques, etc. Therefore, I warmly recommend this monograph not only to those interested in the current topic of stochastic recurrence equations, but also to those who want to learn some modern methods of probability theory." (Norbert Bogya, Acta Scientiarum Mathematicarum, Vol. 83 (1-2), 2017)"It consists of five sections, five appendixes, a list of abbreviations and symbols, 262 references, and an index. It is a well-written and interesting book, and represents a good material for students and researchers." (Miroslav M. Risti, zbMATH 1357.60004, 2017) Review Quote "It consists of five sections, five appendixes, a list of abbreviations and symbols, 262 references, and an index. It is a well-written and interesting book, and represents a good material for students and researchers." (Miroslav M. Ristic, zbMATH 1357.60004, 2017) Feature Covers fields which are not available in book form and are spread over the literature Provides an accessible introduction to a complicated stochastic model A readable overview of one of the most complicated topics on applied probability theory Details ISBN3319806246 Author Thomas Mikosch Year 2018 ISBN-10 3319806246 ISBN-13 9783319806242 Format Paperback DEWEY 330.015195 Pages 320 Publisher Springer International Publishing AG Series Springer Series in Operations Research and Financial Engineering Publication Date 2018-05-30 Imprint Springer International Publishing AG Place of Publication Cham Country of Publication Switzerland Subtitle The Equation X = AX + B Short Title Stochastic Models with Power-Law Tails Language English UK Release Date 2018-05-30 Edition Description Softcover reprint of the original 1st ed. 2016 Alternative 9783319296784 Audience Professional & Vocational Illustrations 5 Illustrations, color; 4 Illustrations, black and white; XV, 320 p. 9 illus., 5 illus. in color. 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