On stationary Markov chains and independent random variables

Abstract

Two new proofs are given for the fact that a stationary, irreducible, aperiodic Markov chain ( X n n = …, −1,0,1,2…) with denumerable state space has a representation of the form X′ n=g(U n−1, U n−2,…) , where g is a measurable function, ( U n , n= …, −1,0,1,2,…) a sequence of independent random variables uniformly distributed on (0,1), and ( X′ n ) has the same probability law as ( X n ).