Majorizing Potentials in Strong Ratio Limit Theorems

In [1], the strong ratio limit theorems associated with Markov chains were first proved for some “test” functions with specific properties and were then generalized to a wider family of functions. In the present paper, this family is significantly extended by functions that can be majorized in a sense by the potentials of the original functions. The verification of whether a function belongs of the new family can be simplified by using small functions and their analogs. Here the traditional recurrency- or irreducibility-type requirements for the corresponding Markov chains are replaced by more flexible requirements.