Absorption Probabilities for Sums of Random Variables Defined on a Finite Markov Chain

Abstract

Summary This paper is essentially a continuation of the previous one (5) and the notation established therein will be freely repeated. The sequence {ξ r } of random variables is defined on a positively regular finite Markov chain { k r } as in (5) and the partial sums and are considered. Let ζ n be the first positive ζ r and let π jk (y) , the ‘ruin’ function or absorption probability, be defined by The main result (Theorem 1) is an asymptotic expression for π jk (y) for large y in the case when , the expectation of ξ 1 being computed under the unique stationary distribution for k 0 , the initial state of the chain, and unconditional on k 1 .