A Bound for Norms in Lp(T) of Deviations of φ-sub-Gaussian Stochastic Processes

Abstract

We study deviations of ϕ-sub-Gaussian stochastic process from a measurable function and generalize the results of [6]. We obtain a bound for the distributions of norms in the space L p (T). As an example, the obtained result is applied for an aggregate of independent processes of generalized ϕ-sub-Gaussian fractional Brownian motions.MSC: 60G07, 60G18

On Supremum Distribution of Averaged Deviations of Random Orlicz Processes from the Class 𝚫𝟐

“…Similar problems considered for different classes of Orlicz processes were investigated by many authors (Kozachenko and Ryazantseva, 1992;Kozachenko and Moklyachuk, 2003;Kozachenko et al, 2005;Kozachenko and Sergiienko, 2014;Kozachenko and Mlavets, 2015;Yamnenko, 2015Yamnenko, , 2016.…”

“…Kozachenko and Mlavets (2015) estimated the accuracy and reliability (in Lp(T) metrics) for the calculation of improper integrals depending on a parameter using the Monte Carlo method and the theory of Orlicz subspace Fψ(Ω). Yamnenko (2015) studied deviations of stochastic processes from Orlicz spaces of exponential type and obtained a bound for the distributions of norms in the space Lp(T). Yamnenko (2016) obtained an estimate for distributions of norms of deviations of a stochastic process from the Orlicz space from the class Δ2 of Orlicz functions.…”

Averaged deviations of Orlicz processes and majorizing measures

“…Particular cases of problems considered in this paper were investigated by Kozachenko and Moklyachuk [7], Kozachenko and Ryazantseva [8], Kozachenko, Vasylyk, and Yamnenko [10], Kozachenko and Sergiienko [9], Yamnenko [18]. Kozachenko and Ryazantseva [8] obtained conditions of boundedness and sample path continuity with probability one of stochastic processes from the Orlicz space of random variables generated by exponential Orlicz functions.…”

“…Kozachenko and Sergiienko [9] constructed tests for a hypothesis concerning the form of the covariance function of a Gaussian stochastic process. Yamnenko [18] obtained an estimate for distributions of norms of deviations of a stochastic process from the Orlicz space of exponential type from a given function in L p (T).…”