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Random Operators and Stochastic Equations

Editor-in-Chief: Girko, Vyacheslav

Managing Editor: Molchanov, S.

Editorial Board: Accardi, L. / Albeverio, Sergio / Carmona, R. / Casati, G. / Christopeit, N. / Domanski, C. / Drygas, Hilmar / Gupta, A.K. / Ibragimov, I. / Kirsch, Werner / Klein, A. / Kondratyev, Yuri / Kurotschka, V. / Leonenko, N. / Loubaton, Philippe / Orsingher, E. / Pastur, L. / Rodrigues, Waldyr A. / Shiryaev, Albert / Turbin, A.F. / Veretennikov, Alexandre

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Volume 23, Issue 3


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

Rostyslav E. Yamnenko
Published Online: 2015-08-13 | DOI: https://doi.org/10.1515/rose-2014-0039


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 Lp(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.

Keywords: Sub-Gaussian random variable; sub-Gaussian random process; Orlicz space; deviation of random process; Wiener process

MSC: 60G07; 60G15

About the article

Received: 2014-12-17

Accepted: 2015-06-04

Published Online: 2015-08-13

Published in Print: 2015-09-01

Citation Information: Random Operators and Stochastic Equations, Volume 23, Issue 3, Pages 187–194, ISSN (Online) 1569-397X, ISSN (Print) 0926-6364, DOI: https://doi.org/10.1515/rose-2014-0039.

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