Boundary-value problems for sums of random variables, defined on a countably-valued Markov chain. II
[For part I see the preceding entry, Zbl 0699.60068.] Let \((\xi_ n,x_ n)\) be a Markov additive process where \(x_ n\) is an infinite Markov chain and \(\xi_ n-\xi_{n-1}\) takes values in the set of integers. Assume that \[ P\{\xi_ 1-\xi_ 0\geq -1| x_ 0=i\}=1\quad and\quad P\{\xi_ 1-\xi_ 0=-1| x_ 0=i_ 0\}>0\quad for\quad some\quad i_ 0. \] Set \(\tau_ k=\inf \{n: \xi_ n>k\}\), \(\eta_ k=\xi_{\tau_ k}-k\). The main results of the article are connected with the asymptotic behaviour of the functions \[ (E\{e^{- \lambda \tau_ k/k^ 2},x_{\tau_ k}=j| x_ 0=i\}),\quad and\quad (P\{\eta_ K>\ell,x_{\tau_ k}=j| x_ 0=i\}) \] when \(k\to \infty\).
scientific article; zbMATH DE number 4147249