##
**Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness.**
*(English)*
Zbl 0983.60005

Lecture Notes in Mathematics. 1766. Berlin: Springer. viii, 145 p. (2001).

It is well-known that the techniques from the perturbation theory of operators, applied to a quasi-compact positive kernel \(Q,\) for obtaining limit theorems for Markov chains or for describing stochastic properties of dynamical systems, by use of a Perron-Frobenius operator, has been demonstrated in several papers [see, for example, J.-P. Conze and A. Raugi, “Convergence des potentiels pour un opérateur de transfert, applications aux systèmes dynamiques et aux chaînes de Markov” (2000; Zbl 0983.60071)].

In this book the authors give a general functional analytical framework for the method of quasi-compactness and prove the asymptotic behaviour within it. The main part of the book deals with a quasi-compact Markov kernel \(Q\) for which \(1\) is a simple eigenvalue but is not the unique eigenvalue of modulus \(1.\) An essential element of the work is the precise description of the peripheral spectrum of \(Q\) and of its perturbations. The main aim of the book is to establish limit theorems for the sequence of r.v. \((\xi(X_{n}))_{n\geq 0}, \) namely, CLT, renewal theorems, large deviations theorems, where \(\xi\) is a measurable, real-valued function on a measurable space \((E, {\mathcal E})\) and \((X_{n})_{n\geq 0}\) is a Markov chain on \((E,{\mathcal E})\) associated with a transition probability \(Q.\) Also, the authors study some stochastic properties of dynamical systems: (a) existence of so-called \(\tau\)-invariant probability distribution; (b) asymptotical properties of r.v.s \((\xi\circ\tau^{k})_{k\geq 0}.\) It may be treated in the context of a quasi-compact action of the positive non-Markov kernel \(Q\) on a Banach space. The obtained results are extended to kernels for which \(1\) is an eigenvalue of multiplicity greater than one.

The only prerequisite for this book is a knowledge of the basic techniques of probability theory and of notions of elementary functional analysis. All the proofs are complete and not complicated. This book may be useful for a better understanding and new uses of the famous methods.

In this book the authors give a general functional analytical framework for the method of quasi-compactness and prove the asymptotic behaviour within it. The main part of the book deals with a quasi-compact Markov kernel \(Q\) for which \(1\) is a simple eigenvalue but is not the unique eigenvalue of modulus \(1.\) An essential element of the work is the precise description of the peripheral spectrum of \(Q\) and of its perturbations. The main aim of the book is to establish limit theorems for the sequence of r.v. \((\xi(X_{n}))_{n\geq 0}, \) namely, CLT, renewal theorems, large deviations theorems, where \(\xi\) is a measurable, real-valued function on a measurable space \((E, {\mathcal E})\) and \((X_{n})_{n\geq 0}\) is a Markov chain on \((E,{\mathcal E})\) associated with a transition probability \(Q.\) Also, the authors study some stochastic properties of dynamical systems: (a) existence of so-called \(\tau\)-invariant probability distribution; (b) asymptotical properties of r.v.s \((\xi\circ\tau^{k})_{k\geq 0}.\) It may be treated in the context of a quasi-compact action of the positive non-Markov kernel \(Q\) on a Banach space. The obtained results are extended to kernels for which \(1\) is an eigenvalue of multiplicity greater than one.

The only prerequisite for this book is a knowledge of the basic techniques of probability theory and of notions of elementary functional analysis. All the proofs are complete and not complicated. This book may be useful for a better understanding and new uses of the famous methods.

Reviewer: Anatoly Swishchuk (Kyïv)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60F10 | Large deviations |

60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |

37A25 | Ergodicity, mixing, rates of mixing |

37A50 | Dynamical systems and their relations with probability theory and stochastic processes |

60F05 | Central limit and other weak theorems |