Conditions That a Stochastic Process be Ergodic

Abstract

This note gives necessary and sufficient conditions in termsof characteristic functions that a strictly stationary stochasticprocess $X(t)$ be metrically transitive or ergodic. A mean ergodictheorem is stated for stochastic processes which are strictly stationaryof order K, by which is meant that for every choice of $K$ points$t_1,…,t_k$, the random variables $X(t1+h),…,X(tk+h)$ have ajoint probability distribution which does not depend on $h$