Some asymptotic results for the semi-Markovian random walk with a special barrier

Abstract

In this paper, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered. Under the assumption that the random variables {zeta(n)}, n >= 1 describing discrete interference of chance are in the form of an ergodic Markov chain with Weibull stationary distribution, the ergodic theorem for the process X(t) is proved. By using basic identity, the characteristic function of the process X(t) is expressed by the characteristics of a boundary functional S-N(x). Moreover, the asymptotic expansions with three terms for the first four moments of the ergodic distribution of the process X(t) are obtained, when the expected value of the jump at time of discrete interference of chance tends to infinity.

[Aliyev, Rovshan] Baku State Univ, Dept Probabil Theory & Math Stat, Baku, Azerbaijan; [Aliyev, Rovshan; Kesemen, Tulay] Karadeniz Tech Univ, Dept Math, TR-61080 Trabzon, Turkey; [Khaniyev, Tahir] TOBB Univ Econ & Technol, Dept Ind Engn, Ankara, Turkey; [Khaniyev, Tahir] Azerbaijan Natl Acad Sci, Inst Cybernet, Baku, Azerbaijan