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**Mixing: Properties and examples.**
*(English)*
Zbl 0801.60027

Lecture Notes in Statistics (Springer). 85. New York: Springer-Verlag. xii, 142 p. (1994).

The components of some random processes or fields are in a certain sense weakly dependent. One of the different ways to measure this kind of dependence is introducing the so-called mixing coefficients. In the first half of the present monograph the general properties of these coefficients are investigated. The coefficients in question are the strong mixing \((\alpha\)-mixing), the uniform mixing \((\varphi\)-mixing), the \(*\)-mixing \((\psi\)-mixing) and the maximal correlation \((\rho\)- mixing) coefficients and the coefficient of absolute regularity \((\beta\)- mixing coef.). The author gives a short description of the relations between these various coefficients and discusses the difference of mixing for processes and for fields. Then he develops the tools now available for working with mixing processes and fields. These are covariance inequalities, Berbee’s and Bradley’s reconstruction theorems, a Rosenthal-type moment inequality, variations of Hoeffding’s and Bernstein’s exponential inequalities and maximal inequalities. This part of the book is finished by quotation of some results concerning the central limit theorem (for instance dimension dependent convergence rates).

Examples for mixing processes and fields are considered in the second half of the monograph. For discrete Gaussian fields, Gibbs fields, linear fields and Markov chains with general state space the author gives conditions implying convergence of the mixing coefficients at fixed (e.g. geometric) rates. Various models of auto-regressive sequences are treated as special cases of Markov chains. For time-continuous Markov processes, mixing properties are deduced by considering the infinitesimal operators of the processes. Finally the author introduces the concept of hypermixing originating from large deviation theory. In this context also the notions of hyper- and ultracontractivity of Markov semigroups are discussed.

The book represents an overview of the theory of mixing processes and fields which seems to be useful for researchers. In this respect it is a continuation and supplement of E. Eberlein and M. S. Taqqu (eds.) [Dependence in probability and statistics (Prog. Prob. Stat. 11, BirkhĂ¤user, Boston, 1986)]. A lot of the results is proved or the proofs are sketched. Concerning linear fields, in Section 2.3 the proof of the Theorems 1 and 3 which generalize considerably a result of V. V. Gorodetskij [Theory Probab. Appl. 22(1977), 411-413 (1978); translation from Teor. Veroyatn. Primen. 22, 421-423 (1977; Zbl 0377.60046)] does not seem to be complete. Detailed references are given for all treated and even some omitted subjects.

Examples for mixing processes and fields are considered in the second half of the monograph. For discrete Gaussian fields, Gibbs fields, linear fields and Markov chains with general state space the author gives conditions implying convergence of the mixing coefficients at fixed (e.g. geometric) rates. Various models of auto-regressive sequences are treated as special cases of Markov chains. For time-continuous Markov processes, mixing properties are deduced by considering the infinitesimal operators of the processes. Finally the author introduces the concept of hypermixing originating from large deviation theory. In this context also the notions of hyper- and ultracontractivity of Markov semigroups are discussed.

The book represents an overview of the theory of mixing processes and fields which seems to be useful for researchers. In this respect it is a continuation and supplement of E. Eberlein and M. S. Taqqu (eds.) [Dependence in probability and statistics (Prog. Prob. Stat. 11, BirkhĂ¤user, Boston, 1986)]. A lot of the results is proved or the proofs are sketched. Concerning linear fields, in Section 2.3 the proof of the Theorems 1 and 3 which generalize considerably a result of V. V. Gorodetskij [Theory Probab. Appl. 22(1977), 411-413 (1978); translation from Teor. Veroyatn. Primen. 22, 421-423 (1977; Zbl 0377.60046)] does not seem to be complete. Detailed references are given for all treated and even some omitted subjects.

Reviewer: D.Tasche (Berlin)

### MSC:

60G05 | Foundations of stochastic processes |

60F05 | Central limit and other weak theorems |

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

60F10 | Large deviations |