A Uniform Theory for Sums of Markov Chain Transition Probabilities

Necessary and sufficient conditions are given for boundedness of ₙ \|∑ⁿₖ₌₁ (Pᵏ(x, •) - Pᵏ(y, •))\| and ₙ \|∑ⁿₖ₌₁ (Pᵏ(x, •) - π\|, where the norm is total variation and where π is an invariant probability measure. Also conditions for convergence of ∑^∞ₖ₌₁ (Pᵏ(x, •) - π) in norm are given. These results require the study of certain "small sets." Two new types are introduced: uniform sets and strongly uniform sets, and these are related to the sets introduced by Harris in his study of the existence of σ-finite invariant measure.