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Ergodicity and stability of stochastic processes. Transl. from the Russian by V. Yurinsky. (English) Zbl 0917.60005

Wiley Series in Probability and Statistics. Chichester: Wiley. xxiii, 585 p. (1998).
The primary object of study of the monograph are discrete-time Markov chains with general state space and – more generally – sequences defined by a recursion \(X(n+1)=f(X(n),\xi_n)\), where \(\{\xi_n\}\) is stationary. The author calls such a process a stochastically recursive sequence (SRS) – some other authors use the term random transformation or (discrete-time) random dynamical system. Yet more generally recursive chains (RC) are considered for which the conditional law of \(X(n+1)\) given the past of \(X\) and \(\xi\) equals that given \(X(n)\) and \(\xi_n\). The questions treated are criteria for various ergodicity properties of such processes, e.g. convergence of the law of \(X(n)\) to an invariant probability measure or so-called (strong) coupling convergence to a stationary process. The term stability in the title refers to continuity properties of the limiting measure or process with respect to changes of parameters. Often stability is obtained as a by-product of an ergodicity result.
The first two chapters treat Harris and non-Harris Markov chains. Sufficient conditions for ergodicity are provided using renovation (or regenerative) events, Lyapunov functions or contraction or monotonicity properties of an underlying transformation. In the third chapter many results are generalized or adapted to SRSs and RCs. In Chapter 4 continuous-time processes are analyzed by embedding a suitable discrete-time process and using results of the previous chapters. Chapter 5-10 treat the special case in which the state space is \(\mathbb{R}^d\) or a nice subset thereof (half-space, octant, strip). In the final chapter those results are applied to different types of polling systems, Jackson networks and random access communication systems.
Even though the literature on ergodic properties of stochastic processes has become far too large to collect all relevant material in one monograph, the book does contain so many results, that it will be a valuable reference for researchers and graduate students interested in the subject. The monograph is well-written and contains many examples and additional explanation. The reader should have some basic knowledge in probability and measure theory.

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J25 Continuous-time Markov processes on general state spaces
60J05 Discrete-time Markov processes on general state spaces
60J27 Continuous-time Markov processes on discrete state spaces
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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