Bounds for the distribution of some functionals of processes with $\varphi $-sub-Gaussian increments

Abstract

Abstract. Bounds for the distribution of some functionals of a stochastic process {X(t), t ∈ T } belonging to the class V (ϕ, ψ) are obtained. An example of the functionals studied in the paper is given bywhere f (t) is a continuous function that can be viewed as a service output rate of a queue formed by the process X(t). For the latter interpretation, the bounds can be viewed as upper estimates for the buffer overflow probabilities with buffer size x > 0. The results obtained in the paper apply to Gaussian stochastic processes. As an example, we show an application for the generalized fractional Brownian motion defined on a finite interval.

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

“…It extends a series of papers [4,5,12,13,14] studying probabilities of exceeding by such processes a level specified by a given function in C(T) spaces. Such problems find an application in queuing and risk theories, for example, for estimating the ruin probability or buffer overflow probability in a model with ϕ-sub-Gaussian input process.…”

“…General theory and various applications of sub-Gaussian, ϕ-sub-Gaussian, and strictly ϕ-sub-Gaussian random variables and processes can be found in the book of Buldygin and Kozachenko [1] and in the papers [2,4,5,7,8,9,12,13,14].…”