On the Moments of a Semi-Markovian Random Walk with Gaussian Distribution of Summands

Abstract

In this article, a semi-Markovian random walk with delay and a discrete interference of chance (X(t)) is considered. It is assumed that the random variables ζ n , n = 1, 2,…, which describe the discrete interference of chance form an ergodic Markov chain with ergodic distribution which is a gamma distribution with parameters (α, λ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as λ → 0. Moreover, by using the Riemann zeta-function, the coefficients of these asymptotic expansions are expressed by means of numerical characteristics of the summands, when the process considered is a semi-Markovian Gaussian random walk with small drift β.