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

Abstract

Summary Let { k r } ( r = 0, 1, 2, …; 1 ≤ k r ≤ h ) be a positively regular, finite Markov chain with transition matrix P = ( p jk ). For each possible transition j → k let g jk (x) (− ∞ ≤ x ≤ ∞) be a given distribution function. The sequence of random variables {ξ r } is defined where ξ r has the distribution g jk (x) if the rth transition takes the chain from state j to state k . It is supposed that each distribution g jk (x) admits a two-sided Laplace-Stieltjes transform in a real t -interval surrounding t = 0. Let P(t) denote the matrix { P jk m jk (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).