The existence of moments for stationary Markov chains

Abstract

We give conditions under which the stationary distribution π of a Markov chain admits moments of the general form ∫ f( x) π( dx), where f is a general function; specific examples include f( x) = x ^r and f( x) = e ^sx . In general the time-dependent moments of the chain then converge to the stationary moments. We show that in special cases this convergence of moments occurs at a geometric rate. The results are applied to random walk on [0, ∞).