This article is cited in 5 papers

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

M. G. Shur

Moscow State Institute of Electronics and Mathematics

Abstract: A simplest ratio limit theorem is obtained for self-adjoint operators in the spaces of L2 type which leave invariant a cone of nonnegative elements. By means of the theorem we establish ratio limit theorems for symmetric Markov chains and symmetric kernels in measurable spaces. In particular, it is shown that for symmetric Harris recurrent Markov chains a result is valid which is an analogue of the known Orey theorem (1961) about discrete recurrent symmetric chains. Similar statements are valid for nonnegative symmetric quasi-Feller kernels on locally compact spaces which are Liouville in a certain sense.

Keywords: ratio limit theorem, self-adjoint operator, Harris recurrent Markov chain, symmetric kernel, quasi-Feller kernel, Liouville kernel.

Received: 14.07.1997

DOI: 10.4213/tvp463

 English version: Theory of Probability and its Applications, 2001, 45:2, 273–288

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