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 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 S_N(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 W_β(t) = 2X(t)/β − 1 as β → ∞ is studied for the case in which the components of the walk (η_i) have a two-sided exponential distribution.