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 (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 (, ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as 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 .

[Aliyev, Rovshan] Baku State Univ, Dept Probabil Theory & Math Stat, Baku, Azerbaijan; [Khaniyev, Tahir] TOBB Univ Econ & Technol, Dept Ind Engn, Ankara, Turkey; [Aliyev, Rovshan; Khaniyev, Tahir] Azerbaijan Natl Acad Sci, Inst Cybernet, Baku, Azerbaijan