##
**\(\varphi\)-sub-Gaussian random processes.
(\(\варпхи\)-субгауссові випадкові процеси.)**
*(Ukrainian)*
Zbl 1224.60070

Kyïv: Vydavnycho-Poligrafichnyĭ Tsentr, Kyïvskyĭ Universytet (ISBN 978-966-439-051-1). 231 p. (2008).

This monograph is devoted to the theory of spaces of \(\varphi\)-sub-Gaussian random variables and \(\varphi\)-sub-Gaussian random processes. The authors mention that many real processes, which are believed to be Gaussian or close to Gaussian, in fact, have nonsymmetric densities of one-dimensional distributions and its distribution “tails” are heavier or easier than Gaussian ones. For example, in the monograph of V. V. Buldygin and Yu. V. Kozachenko [Metric characterization of random variables and random processes. Providence, RI: AMS (2000; Zbl 0998.60503)] various classes of such random variables and processes are studied. The notion of \(\varphi\)-sub-Gaussian random variables generalizes the notion of sub-Gaussian random variables introduced in the works of J. P. Kahane [Stud. Math. 19, 1–25 (1960; Zbl 0096.11402); Some random series of functions. Lexington, Mass.: D.C. Heath and Company, a Division of Raytheon Education Company (1968; Zbl 0192.53801)]. Recall that a centred random variable \(\xi\) is sub-Gaussian if there exists a constant \(a>0\) such that, for all \(\lambda\in\mathbb R\), the following inequality holds true \(\operatorname{E}\exp\{\lambda\xi\}\leq\exp\{\lambda^2a^2/2\},\) i.e., the moment generating function of \(\xi\) is majorized by the moment generating function of a Gaussian random variable. A \(\varphi\)-sub-Gaussian random variable \(\xi\) is a centred random variable for which there exists a constant \(a>0\) such that, for all \(\lambda\in\mathbb R\), the inequality \(\operatorname{E}\exp\{\lambda\xi\}\leq\exp\{\varphi(\lambda a)\}\) holds true, where \(\varphi\) is an Orlicz N-function.

The book consists of 8 chapters. The authors’ description of the chapters gives some impression of the book. Chapter 1 presents all essential definitions and statements from the theory of \(\varphi\)-sub-Gaussian random variables. Random processes from the classes \(V(\varphi,\psi)\) and the spaces \(\text{Sub}_{\varphi}(\Omega)\) are considered in Chapter 2. Conditions for sample paths’ continuity with probability one and distribution estimates for some functionals of such processes are considered in Chapter 2. Conditions for the weak convergence of a family of random processes from the class \(V(\varphi,\psi)\) in the space \(C_0(\mathbb R^+,q)\) and on a compact set are studied in Chapter 3. Properties of strictly \(\varphi\)-sub-Gaussian generalized fractional Brownian motions and queues created by such processes are considered in Chapter 4. Chapter 5 is devoted to the construction of algorithms for the modelling of the \(\varphi\)-sub-Gaussian generalized fractional Brownian motion. A model of a process that approximates it with given reliability and accuracy in the space \(C([0,1])\) is constructed. It is shown in Chapter 6, how the theory of \(\varphi\)-sub-Gaussian random processes can be applied to modelling continuous in mean square sense stationary Gaussian processes with continuous spectral function. Chapter 7 contains conditions for uniform convergence with probability one of the wavelet expansion of a \(\varphi\)-sub-Gaussian random process. It is shown that, under certain conditions on the wavelet basis of a wavelet expansion of a stationary Gaussian process with trajectories that are continuous with probability one, the wavelet expansion is uniformly convergent with probability one on any bounded interval. Chapter 8 studies inequalities of exponential type for the distribution of the supremum of a random field that arises as a solution of the thermal conductivity equation with random initial conditions.

The material is substantially based on the results obtained by the authors and their co-authors. See, for example, [the authors, Random Oper. Stoch. Equ. 13, No. 2, 111–128 (2005; Zbl 1118.60025); the first and the second author and T. Sottinen, Methodol. Comput. Appl. Probab. 7, No. 3, 379–400 (2005; Zbl 1082.60512); the second author and G. M. Leonenko, Extremes 8, No. 3, 191–205 (2005; Zbl 1115.60054); the first and the second author and M. M. Perestyuk, Random Oper. Stoch. Equ. 14, No. 3, 209–232 (2006; Zbl 1120.60036)].

The monograph should be of interest to researchers in the fields of radio engineering, physics of the atmosphere, geophysics, financial mathematics, mathematical economics and others, who use the methods of the theory of random processes.

The book consists of 8 chapters. The authors’ description of the chapters gives some impression of the book. Chapter 1 presents all essential definitions and statements from the theory of \(\varphi\)-sub-Gaussian random variables. Random processes from the classes \(V(\varphi,\psi)\) and the spaces \(\text{Sub}_{\varphi}(\Omega)\) are considered in Chapter 2. Conditions for sample paths’ continuity with probability one and distribution estimates for some functionals of such processes are considered in Chapter 2. Conditions for the weak convergence of a family of random processes from the class \(V(\varphi,\psi)\) in the space \(C_0(\mathbb R^+,q)\) and on a compact set are studied in Chapter 3. Properties of strictly \(\varphi\)-sub-Gaussian generalized fractional Brownian motions and queues created by such processes are considered in Chapter 4. Chapter 5 is devoted to the construction of algorithms for the modelling of the \(\varphi\)-sub-Gaussian generalized fractional Brownian motion. A model of a process that approximates it with given reliability and accuracy in the space \(C([0,1])\) is constructed. It is shown in Chapter 6, how the theory of \(\varphi\)-sub-Gaussian random processes can be applied to modelling continuous in mean square sense stationary Gaussian processes with continuous spectral function. Chapter 7 contains conditions for uniform convergence with probability one of the wavelet expansion of a \(\varphi\)-sub-Gaussian random process. It is shown that, under certain conditions on the wavelet basis of a wavelet expansion of a stationary Gaussian process with trajectories that are continuous with probability one, the wavelet expansion is uniformly convergent with probability one on any bounded interval. Chapter 8 studies inequalities of exponential type for the distribution of the supremum of a random field that arises as a solution of the thermal conductivity equation with random initial conditions.

The material is substantially based on the results obtained by the authors and their co-authors. See, for example, [the authors, Random Oper. Stoch. Equ. 13, No. 2, 111–128 (2005; Zbl 1118.60025); the first and the second author and T. Sottinen, Methodol. Comput. Appl. Probab. 7, No. 3, 379–400 (2005; Zbl 1082.60512); the second author and G. M. Leonenko, Extremes 8, No. 3, 191–205 (2005; Zbl 1115.60054); the first and the second author and M. M. Perestyuk, Random Oper. Stoch. Equ. 14, No. 3, 209–232 (2006; Zbl 1120.60036)].

The monograph should be of interest to researchers in the fields of radio engineering, physics of the atmosphere, geophysics, financial mathematics, mathematical economics and others, who use the methods of the theory of random processes.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

### MSC:

60G07 | General theory of stochastic processes |

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

60G17 | Sample path properties |