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 Teor. Veroyatnost. i Primenen., 2000, Volume 45, Issue 2, Pages 268–288 (Mi tvp463)

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.

DOI: https://doi.org/10.4213/tvp463

Full text: PDF file (1160 kB)

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

Bibliographic databases:

Citation: M. G. Shur, “Ratio limit theorems for self-adjoint operators and symmetric Markov chains”, Teor. Veroyatnost. i Primenen., 45:2 (2000), 268–288; Theory Probab. Appl., 45:2 (2001), 273–288

Citation in format AMSBIB
\Bibitem{Shu00} \by M.~G.~Shur \paper Ratio limit theorems for self-adjoint operators and symmetric Markov chains \jour Teor. Veroyatnost. i Primenen. \yr 2000 \vol 45 \issue 2 \pages 268--288 \mathnet{http://mi.mathnet.ru/tvp463} \crossref{https://doi.org/10.4213/tvp463} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1967757} \zmath{https://zbmath.org/?q=an:0982.60061} \transl \jour Theory Probab. Appl. \yr 2001 \vol 45 \issue 2 \pages 273--288 \crossref{https://doi.org/10.1137/S0040585X97978221} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000169004700007} 

• http://mi.mathnet.ru/eng/tvp463
• https://doi.org/10.4213/tvp463
• http://mi.mathnet.ru/eng/tvp/v45/i2/p268

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. G. Shur, “On the Lin Condition in Strong Ratio Limit Theorems”, Math. Notes, 75:6 (2004), 864–876
2. M. G. Shur, “Majorizing Potentials in Strong Ratio Limit Theorems”, Math. Notes, 84:1 (2008), 116–124
3. M. G. Shur, “Convergence Parameter Associated with a Markov Chain and a Family of Functions”, Math. Notes, 87:2 (2010), 271–280
4. M. G. Shur, “Uniform integrability for strong ratio limit theorems. II”, Theory Probab. Appl., 55:3 (2011), 473–484
5. M. G. Shur, “Two theorems on convergence parameter of an irreducible Markov chain”, Theory Probab. Appl., 58:1 (2014), 159–164
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