On the Limiting Behavior of the Characteristic Function of the Ergodic Distribution of the Semi-Markov Walk With Two Boundaries

Abstract

The semi-Markov walk (X(t)) with two boundaries at the levels 0 and beta > 0 is considered. The characteristic function of the ergodic distribution of the processX(t) is expressed in terms of the characteristics of the boundary functionals N(z) and S (N)(z), where N(z) is the firstmoment of exit of the random walk {Sn}, n >= 1, from the interval (-z, beta - z), z a [0, beta]. The limiting behavior of the characteristic function of the ergodic distribution of the process W (beta) (t) = 2X(t)/beta - 1 as beta -> a is studied for the case in which the components of the walk (eta (i)) have a two-sided exponential distribution.

[Aliyev, R. T.] Baku State Univ, Baku, Azerbaijan; [Aliyev, R. T.; Khaniyev, T. A.] Natl Acad Sci, Guseynov Inst Cybernet, Baku, Azerbaijan; [Khaniyev, T. A.] TOBB Econ & Technol Univ, Ankara, Turkey