A Convexity Property in the Theory of Random Variables Defined on a Finite Markov Chain

1. Summary. Let P = (pjk) be the transition matrix of an ergodic, finite Markov chain with no cyclically moving sub-classes. For each possible transition (j, k), let Hj,k(x) be a distribution function admitting a moment generating function fjk(t) in an interval surrounding t = 0. The matrix P(t) = {pjkfjk(t)} is of interest in the study of the random variable S. = Xi + * * * + X., where Xm has the distribution Hjk(x) if the mth transition takes the chain from state j to state k. The matrix P(t) is non-negative and theref6re possesses a maximal positive eigenvalue ai(t), which is shown to be a convex function of t. As an application of the convexity property, we obtain an asymptotic expression for the probability of tail values of the sum S. Xin the case where the Xm are integral random variables. The results are related to those of Blackwell and Hodges [1], whose methods are followed closely in Section 5, and Volkov [4], [5], who treats in detail the case of integer-valued functions of the state of the chain, i.e., the case fJk(t) = exp(flkt) (fik integral).