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

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 ) = 2 X ( 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.