Averaged deviations of Orlicz processes and majorizing measures

TL;DR

This paper investigates the supremum of averaged deviations of Orlicz processes using majorizing measures, providing distribution estimates and applications to processes in L2 spaces with known covariance functions.

Contribution

It introduces new bounds for the supremum of averaged deviations of Orlicz processes using majorizing measures, including special cases for L_q spaces.

Findings

Derived distribution estimates for supremum of deviations.

Extended results to L_q spaces, including L_2 with known covariance.

Applied findings to specific stochastic processes in L_2 space.

Abstract

This paper is devoted to investigation of supremum of averaged deviations $|X(t)-f(t)-\int_{\mathbb {T}}(X(u)-f(u))\,\mathrm {d}\mu(u)/\mu(\mathbb {T})|$ of a stochastic process from Orlicz space of random variables using the method of majorizing measures. An estimate of distribution of supremum of deviations $|X(t)-f(t)|$ is derived. A special case of the $L_q$ space is considered. As an example, the obtained results are applied to stochastic processes from the $L_2$ space with known covariance functions.