Ratio limit theorems for self-adjoint operators and symmetric Markov chains (Q2752953)

Standard ratio limit theorems for Markov chains on measurable spaces include either excessive conditions or conditions which, despite their closeness to the necessary ones, are difficult to check. On the other hand, an important and elegant \textit{S. Orey} theorem [``Lecture notes on limit theorems for Markov chain transition probabilities'' (London, 1971)] dealing with discrete symmetric Markov chains does not contain conditions of such a kind. The author establishes two theorems of Orey type for symmetric Harris recurrent Markov chains and symmetric quasi-Feller topological Liouville kernels. Similar statements are proved for nonnegative symmetric quasi-Feller kernels on locally compact spaces which are Liouville in a certain sense. The idea of proofs is based on a new simple limit theorem for ratios generated by some self-adjoint operators.