Small and Pseudo-Small Sets for Markov Chains

In this paper we examine the relationship between small sets and their generalisation, pseudo-small sets. We consider conditions which imply the equivalence of the two notions, and give examples where they are definitely different. We give further examples where sets are both pseudo-small and small, but the minorisation constants implied by the two notions are different. Applications of recent computable bounds results are given and extended. We also give a result linking the ideas of monotonicity and minorisation. Specifically we demonstrate that if a non-monotone chain satisfies a minorisation condition, and furthermore is stochastically dominated by a monotone chain which satisfies a Lyapunov drift condition, then a probability construction exists which incorporates both the bounding process and the minorisation condition.