Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes

Olha Hopkalo,Lyudmyla Sakhno

doi:10.15559/21-vmsta171

Abstract

This paper investigates sample paths properties of φ-sub-Gaussian processes by means of entropy methods. Basing on a particular entropy integral, we treat the questions on continuity and the rate of growth of sample paths. The obtained results are then used to investigate the sample paths properties for a particular class of φ-sub-Gaussian processes related to the random heat equation. We derive the estimates for the distribution of suprema of such processes and evaluate their rate of growth.

Highlights

This paper is devoted to the investigation of important classes of exponential type Orlicz spaces of random variables, namely, φ-sub-Gaussian random variables
In [16, 17] similar results were obtained for the case of higher order heattype equations, but under the conditions stated in terms of a different entropy integral
We investigate the rate of growth of φ-sub-Gaussian processes

Summary

Introduction

This paper is devoted to the investigation of important classes of exponential type Orlicz spaces of random variables, namely, φ-sub-Gaussian random variables. The Gaussian and non-Gaussian limiting distributions for the heat equations with singular data are presented in [1, 23], which can be considered as the starting point for numerous further studies in this area In another series of papers, solutions to partial differential equations subject to random initial conditions were investigated by means of Fourier methods, representations of solutions by uniformly convergent series and their approximations in different functional spaces were developed (see, for example, [11, 19, 20] among many others). In [16, 17] similar results were obtained for the case of higher order heattype equations, but under the conditions stated in terms of a different entropy integral (see [16] for more references on the theory of φ-sub-Gaussian processes and additional references on partial differential equations with random initial conditions)

Preliminaries

Properties of sample paths of φ-sub-Gaussian processes

Stochastic processes related to the heat equation

Conclusions and future studies