A Note on Compact Markov Operators

Let (X,P ) be an irreducible, random walk on the state space X which is at most countable. We suppose that the (usually infinite) stochastic matrix P describes a Markov chain {Zn}n∈N defined on a probability space (Ω,F ,P) with transition probabilities p(x, y) := P[Zn+1 = y|Zn = x] homogeneous in time. Besides we consider the n-step transition probabilities {p(x, y)}x,y∈X which represent the stochastic matrix associated to the n-th convolution power of P . The Markov operator associated to the random walk is defined as follows