Geodesic
On the Lin Condition in Strong Ratio Limit Theorems
Matematičeskie zametki, Tome 75 (2004) no. 6, pp. 927-940

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A new version of strong limit theorems for Lin ratios (1976) free of former restrictions such as the absolute continuity of the measures in question or the conservativeness of the corresponding operators is presented. As the most original statement of the paper (Theorem 5) states, one of the two basic Lin conditions holds automatically for a wide class of Markov chains and, in particular, for many random walks on groups. The proof is based on a substantial development of the author"s results (1980) concerning the behavior of multistep transition probabilities. 
@article{MZM_2004_75_6_a10,
     author = {M. G. Shur},
     title = {On the {Lin} {Condition} in {Strong} {Ratio} {Limit} {Theorems}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {927--940},
     publisher = {mathdoc},
     volume = {75},
     number = {6},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_75_6_a10/}
}
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M. G. Shur. On the Lin Condition in Strong Ratio Limit Theorems. Matematičeskie zametki, Tome 75 (2004) no. 6, pp. 927-940. http://geodesic.mathdoc.fr/item/MZM_2004_75_6_a10/