1 - The Distribution of the Estimates for the Norm of Sub-Gaussian Stochastic Processes

Abstract

This chapter is devoted to the study of the conditions and rate of convergence of sub-Gaussian random series in some Banach spaces. The results of this chapter are used in other chapters to construct the models of Gaussian random processes that approximate them with specified reliability and accuracy in a certain functional space. Generally, the Gaussian stochastic processes are considered, which can be represented as a series of independent items. It should be noted that, as will be shown, these models will not always be Gaussian random processes. The Gaussian models of stationary processes are sub-Gaussian processes. The accuracy of simulation is studied in the spaces C(T), Lp(T), p>0, and Orlicz space LU(T), where T is a compact (usually segment) and U is some C-function. In addition, these models can be used to construct the models of sub-Gaussian processes that approximate them with a given reliability and accuracy in a case when the process can be performed as a sub-Gaussian series with independent items. Provides the necessary information from the theory of the sub-Gaussian random variables space. Sub-Gaussian random variables were introduced for the first time by Kahane. Buldygin and Kozachenko in their publication showed that the space of sub-Gaussian random variables is Banach space. The properties of this space are studied in the work of Buldygin and Kozachenko. Deals with necessary properties of the theory for strictly sub-Gaussian random variables. In, this theory is described in more detail. Note that a Gaussian centered random variable is strictly sub-Gaussian. Therefore, all results of this section, as well as other results of this book, obtained for sub-Gaussian random variables and processes are also true for the centered Gaussian random variables and processes. In the rate of convergence of sub-Gaussian random series in the space L2(T) is found. Similar results are contained. Looks at the distribution estimate of the norm of sub-Gaussian random processes in space Lp (T). These estimates are also considered. For more general spaces, namely the spaces Subφ(Ω) such estimates can also be found. These estimates are used to find the rate of convergence of sub-Gaussian functional series in the norm of spaces Lp(Ω). Note that in the case where p=2, the results are better than. In the estimates of distribution of the sub-Gaussian random processes norm in some Orlicz spaces are found; in these estimates are used to obtain the rate of convergence of sub-Gaussian random series in the norm of some Orlicz spaces.