New $L_1$ and pointwise limit theorems are proved for the transition functions $P^n(x, E)$ of a Markov process with $\sigma$-finite invariant measure $\pi$ satisfying a recurrence condition. Also given are related results about the operators on functions and measures induced by these transition functions. The method depends upon the application of martingale theorems, and the principal restriction concerns the structure of a certain $\sigma$-field.
Richard Isaac. "Limit Theorems for Markov Transition Functions." Ann. Math. Statist. 43 (2) 621 - 626, April, 1972. https://doi.org/10.1214/aoms/1177692641