A Matrix Factorization Problem in the Theory of Random Variables Defined on a Finite Markov Chain

Abstract

SummaryLet {kr} (r = 0, 1, 2, …; 1 ≤ kr ≤ h) be a positively regular, finite Markov chain with transition matrix P = (pjk). For each possible transition j → k let gjk(x)(− ∞ ≤ x ≤ ∞) be a given distribution function. The sequence of random variables {ξr} is defined where ξr has the distribution gjk(x) if the rth transition takes the chain from state j to state k. It is supposed that each distribution gjk(x) admits a two-sided Laplace-Stieltjes transform in a real t-interval surrounding t = 0. Let P(t) denote the matrix {Pjkmjk(t)}. It is shown, using probability arguments, that I − sP(t) admits a Wiener-Hopf type of factorization in two ways for suitable values of s where the plus-factors are non-singular, bounded and have regular elements in a right half of the complex t-plane and the minus-factors have similar properties in an overlapping left half-plane (Theorem 1).