On distribution of the norm of deviation of a sub-Gaussian random process in Orlicz spaces

Abstract

The paper is devoted to the study of sub-Gaussian random variables and stochastic processes. Recall that along with centered Gaussian random variables the space Sub(Ω) of sub-Gaussian random variables contains all bounded zero-mean random variables and all zero-mean random variables whose distribution tails decrease no slower than the tails of the distribution of a Gaussian random variable. Here we study a square deviation of a sub-Gaussian random process from a constant and derive an upper estimate for the exponential moment of the deviation. The obtained result allows to estimate the distribution of deviation of a sub-Gaussian random process from some measurable function in the norm of L_p(𝕋) and in the norm of Orlicz space. The paper generalizes results of [Theory Probab. Math. Statist. 58 (1999), 51–66] for the norm of a sub-Gaussian random process in Orlicz space. As an example we apply the obtained estimates to a sub-Gaussian Wiener process deviated from a linear and a square root functions.