Characterization and sufficient conditions for normed ergodicity of Markov chains

Abstract

Normed ergodicity is a type of strong ergodicity for which convergence of the nth step transition operator to the stationary operator holds in the operator norm. We derive a new characterization of normed ergodicity and we clarify its relation with exponential ergodicity. The existence of a Lyapunov function together with two conditions on the uniform integrability of the increments of the Markov chain is shown to be a sufficient condition for normed ergodicity. Conversely, the sufficient conditions are also almost necessary.