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**Metric characteristics of random variables and processes.
(Metricheskie kharakteristiki sluchajnykh velichin i protsessov.)**
*(Russian)*
Zbl 0933.60031

Teoriya Veroyatnostej i Matematicheskaya Statistika/Probability Theory and Mathematical Statistics. 3. Kiev: TViMS. 290 p. (1998).

The metric and the entropy methods of investigation of random processes are exploited. The authors consider random processes such that their values or increments belong to certain spaces of random variables. They consider \(K_{\sigma}\)-spaces and Orlicz spaces of random variables, spaces of sub-Gaussian, pre-Gaussian, square-Gaussian random variables and other special spaces.

The monograph contains eight chapters. The first chapter deals with spaces of sub-Gaussian and pre-Gaussian random variables. Characteristics of these random variables such as the sub-Gaussian standard, moments and semi-invariant functionals are presented. Large deviation inequalities for random variables and for sums of independent random variables are presented, too. In the second chapter spaces of random variables such as \(K_{\sigma}\)-spaces and Orlicz spaces are investigated. Majorizing characteristics of these spaces are introduced. These characteristics allow to obtain inequalities for maxima of the finite numbers of random variables from these spaces. Exponential Orlicz spaces are considered and large deviations inequalities are proposed. The following two chapters deal with the investigation of analytic properties of random processes from \(K_{\sigma}\)-spaces of random variables. In the third chapter the authors present the condition of boundedness of random processes from \(K_{\sigma}\)-spaces and find estimates for the distribution of suprema of processes from Orlicz spaces. These estimates are strengthened for random processes from Sub\(_{\varphi}(\Omega)\) spaces. Conditions for sample path continuity of random processes from \(K_{\sigma}\)-spaces are presented, too. Moduli of continuity of random processes from the Orlicz space generated by an exponential Orlicz function are studied in this chapter. The authors find the distribution of norms of these random processes in Lipschitz spaces. In the fourth chapter properties of pre-Gaussian processes are studied in more detail.

The following chapters are devoted to applications of results of the previous chapters. In the fifth chapter properties of the generalized processes of Shortky effect are investigated. In the sixth chapter estimates for the correlation function of a Gaussian random process are proposed. The functional tolerance interval for correlation functions of Gaussian processes are constructed. The seventh chapter is devoted to applications of strong sub-Gaussian random processes. The Fourier method of investigation for boundary-value problems with strong sub-Gaussian initial conditions is exploited. The Lévy-Baxter type theorems for strong sub-Gaussian and pseudo sub-Gaussian random processes are proved.

This is a very clear and nicely written monograph. It will be useful to read it even for a nonspecialist of this field.

The monograph contains eight chapters. The first chapter deals with spaces of sub-Gaussian and pre-Gaussian random variables. Characteristics of these random variables such as the sub-Gaussian standard, moments and semi-invariant functionals are presented. Large deviation inequalities for random variables and for sums of independent random variables are presented, too. In the second chapter spaces of random variables such as \(K_{\sigma}\)-spaces and Orlicz spaces are investigated. Majorizing characteristics of these spaces are introduced. These characteristics allow to obtain inequalities for maxima of the finite numbers of random variables from these spaces. Exponential Orlicz spaces are considered and large deviations inequalities are proposed. The following two chapters deal with the investigation of analytic properties of random processes from \(K_{\sigma}\)-spaces of random variables. In the third chapter the authors present the condition of boundedness of random processes from \(K_{\sigma}\)-spaces and find estimates for the distribution of suprema of processes from Orlicz spaces. These estimates are strengthened for random processes from Sub\(_{\varphi}(\Omega)\) spaces. Conditions for sample path continuity of random processes from \(K_{\sigma}\)-spaces are presented, too. Moduli of continuity of random processes from the Orlicz space generated by an exponential Orlicz function are studied in this chapter. The authors find the distribution of norms of these random processes in Lipschitz spaces. In the fourth chapter properties of pre-Gaussian processes are studied in more detail.

The following chapters are devoted to applications of results of the previous chapters. In the fifth chapter properties of the generalized processes of Shortky effect are investigated. In the sixth chapter estimates for the correlation function of a Gaussian random process are proposed. The functional tolerance interval for correlation functions of Gaussian processes are constructed. The seventh chapter is devoted to applications of strong sub-Gaussian random processes. The Fourier method of investigation for boundary-value problems with strong sub-Gaussian initial conditions is exploited. The Lévy-Baxter type theorems for strong sub-Gaussian and pseudo sub-Gaussian random processes are proved.

This is a very clear and nicely written monograph. It will be useful to read it even for a nonspecialist of this field.

Reviewer: M.P.Moklyachuk (Kyïv)

### MSC:

60G07 | General theory of stochastic processes |

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

60G17 | Sample path properties |