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On the limiting behavior of the characteristic function of the ergodic distribution of the semi-Markov walk with two boundaries

Research paper by R. T. Aliyev, T. A. Khaniyev

Indexed on: 02 Dec '17Published on: 23 Oct '17Published in: Mathematical Notes



Abstract

The semi-Markov walk (X(t)) with two boundaries at the levels 0 and β > 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 SN(z), where N(z) is the firstmoment of exit of the random walk {Sn}, n ≥ 1, from the interval (−z, β − z), z ∈ [0, β]. The limiting behavior of the characteristic function of the ergodic distribution of the process Wβ(t) = 2X(t)/β − 1 as β → ∞ is studied for the case in which the components of the walk (ηi) have a two-sided exponential distribution.