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© 2016

Stochastic Models with Power-Law Tails

The Equation X = AX + B

Book

Table of contents

  1. Front Matter
    Pages i-xv
  2. Dariusz Buraczewski, Ewa Damek, Thomas Mikosch
    Pages 1-8
  3. Dariusz Buraczewski, Ewa Damek, Thomas Mikosch
    Pages 9-77
  4. Dariusz Buraczewski, Ewa Damek, Thomas Mikosch
    Pages 79-135
  5. Dariusz Buraczewski, Ewa Damek, Thomas Mikosch
    Pages 137-219
  6. Dariusz Buraczewski, Ewa Damek, Thomas Mikosch
    Pages 221-265
  7. Back Matter
    Pages 267-320

About this book

Introduction

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, Guivarc'h, 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.

Keywords

Power Law Tail Random Iterative Function System Stochastic Recurrence Equation Regular Variation Kesten-Goldie Theory Fixed Point Equation Extreme Value Theory Markov Chain Harmonic Analysis

Authors and affiliations

  1. 1.Institute of MathematicsUniv of WroclawWroclawPoland
  2. 2.Institute of MathematicsUniversity of WroclawWroclawPoland
  3. 3.Department of MathematicsUniversity of CopenhagenCopenhagenDenmark

Bibliographic information

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Reviews

“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. Ristić, zbMATH 1357.60004, 2017)