Coupling, stationarity, and regeneration. (English) Zbl 0949.60007

Thorisson, Hermann

Probability and Its Applications. New York, NY: Springer. xiv, 516 p. DM 159.00; öS 1161.00; sFr 144.00; £55.00; $ 79.95 (2000).

This book is an excellent self-contained exposition of the theory of coupling and its applications. Recall the simplest of the notions of coupling. For each \(i\) in an index set \({\mathbf I}\), let \(Y_i\) be a random element defined on a probability space \((\Omega_i, {\mathcal F}_i, P_i)\) and taking values in a measurable space \((E_i,{\mathcal E}_i)\). A family of random elements \(({\widehat Y}_i, i \in {\mathbf I})\), defined on a common probability space \(({\widehat {\Omega}}, {\widehat {\mathcal F}}, {\widehat P})\), is a coupling of \(Y_i, i \in {\mathbf I}\), if \({\widehat Y}_i \overset{\mathcal D}= Y_i\) (i.e. have equal distributions) for each \(i\). There always exists an independence coupling consisting of independent \({\widehat Y}_i, i \in {\mathbf I}\). In fact, a deep idea and a powerful method is to construct instead of \(Y_i, i \in {\mathbf I}\), a family of their copies \({\widehat Y}_i\), “living” on a common probability space, so that the joint distributions have certain nice properties.
The book is divided into 10 chapters. Chapters 1-2 provide an introduction to the main ideas on coupling for real-valued random variables (e.g., the Poisson approximation for sums of independent 0-1 variables). An interesting example (related to the Bell inequality) deals with measurement of polarization of particles in quantum physics. Markov chains, random walks and elements of the renewal theory are discussed as well (e.g., the classical Blackwell theorem).
A general coupling theory is developed in the next four chapters. In Chapter 3 the extensions of the underlying probability space are considered, namely the techniques of conditioning, transfer and splitting (e.g. transfer is used to obtain Dudley’s extension of the Skorokhod coupling). In Chapter 4 a term exact coupling is introduced for one-sided stochastic processes which coincide from a random coupling time \(T\) onward. In the literature one sometimes calls such processes either coupling or merging. Two generalizations of exact coupling are studied in Chapter 5. Shift-coupling means that the trajectories eventually do not merge “exactly” but only modulo a random time shift. Epsilon-coupling arises in the continuous time case when the random shift can be made arbitrary small. For these three kinds of coupling of processes the same problems are studied. In particular, related distributional versions of the coupling concepts are given, coupling (time) inequalities leading to limiting theory are proved, equivalence of the coupling to an inherent mode of total variation convergence and to coincidence of distributions over an appropriate \(\sigma\)-algebra is established. In Chapter 6 the previous results are applied to Markov processes. Each of the three sets of coupling equivalences is complemented by new statements on triviality, on mixing, on convergence in the state space and on constancy of harmonic functions. Whereas the preceding analysis involved shifting one-sided processes, \(\theta_t Z = (Z_{t+s})_{0 \leq s < \infty}\), Chapter 7 extends the scope: general random elements are viewed and a semigroup of transformations replaces the shift maps \(\theta_t\), \(0 \leq t < \infty\). Thus transformation coupling emerges; it applies, e.g., to random fields indexed by \([0,\infty)^d\).
In the second part of the book coupling appears as a tool. Chapter 8 devoted to relation of stationary and cycle-stationary stochastic processes establishes two Palm dualities between them; the processes being split into cycles by a sequence of random times (points). A shift-coupling interpretation of the duality is given. Chapter 9 extends this theory to random fields associated with a simple point process in \({\mathbb{R}}^d\).
The largest Chapter 10 (of 141 pages) treats regenerative processes generalizing Markov chains and renewal processes. Besides classical regenerative processes behaving like a recurrent Markov chain at the times of visits to a fixed reference state, wide-sense regeneration (an analogue of the Harris chains), time-inhomogeneous regeneration (as in time-inhomogeneous Markov chain with a recurrent state) and taboo regeneration (as in transient Markov chains) are considered. The main topics here are coupling, stationarity and total variation asymptotics.
The whole presentation relies only on basic measure-theoretic probability. As pointed out by the author, the book “is written with Ph.D. students in mind”. However, the first two chapters can be read at the master’s level and even at the advanced undergraduate level. It is a pleasure to mention that the author was awarded the Ólafur Daniélsson Prize in Mathematics for his research, most of which can be found in some form in this book. Certainly, this monograph will be useful for a wide audience of probabilists and statisticians interested in development and employment of methods of modern probability. It can be recommended for using in university courses on stochastic processes and their applications.

Reviewer: Alexander V.Bulinskij (Moskva)

MSC:

60-02	Research exposition (monographs, survey articles) pertaining to probability theory
60Gxx	Stochastic processes
60Jxx	Markov processes
60Kxx	Special processes
60D05	Geometric probability and stochastic geometry
60B15	Probability measures on groups or semigroups, Fourier transforms, factorization
60A10	Probabilistic measure theory
60F99	Limit theorems in probability theory

Keywords:

coupling; stationarity; regeneration; stochastic processes; random fields