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 Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 3, Pages 517–532 (Mi tvp92)

Uniform integrability condition in strong ration limit theorems

M. G. Shur

Moscow State Institute of Electronics and Mathematics

Abstract: For a given Markov chain with a measurable state space $(E,\mathscr{E})$, transition operator $P$, and fixed measurable function $f\geq 0$, under necessary conditions, we consider variables $\mu(f_n)$, where $n\ge 1$ is sufficiently large, $f_n=P^nf/\nu(P^nf)$, and $\mu$ and $\nu$ are probability measures on $\mathscr{E}$. For a wide class of situations we propose sufficient and often necessary and sufficient conditions for the convergence of $f_n$ to 1 as $n\to\infty$. These results differ from the results of Orey, Lin, Nummelin, and others by replacing the traditional recurrent conditions of a chain or the uniform boundedness of the functions $f_n$ and the minorizing condition of [E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators, Cambridge University Press, Cambridge, UK, 1984] with more flexible assumptions, among which the uniform integrability of functions $f_n$ with respect to some collection of measures plays a particular role. Our theorems imply a weak and often a strong convergence of these functions to $\varphi\equiv 1$ in respective spaces of a summable function.

Keywords: Markov chain, strong limit theorem for ratios.

DOI: https://doi.org/10.4213/tvp92

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English version:
Theory of Probability and its Applications, 2006, 50:3, 436–447

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Revised: 15.02.2005

Citation: M. G. Shur, “Uniform integrability condition in strong ration limit theorems”, Teor. Veroyatnost. i Primenen., 50:3 (2005), 517–532; Theory Probab. Appl., 50:3 (2006), 436–447

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp92
• https://doi.org/10.4213/tvp92
• http://mi.mathnet.ru/eng/tvp/v50/i3/p517

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This publication is cited in the following articles:
1. M. G. Shur, “Majorizing Potentials in Strong Ratio Limit Theorems”, Math. Notes, 84:1 (2008), 116–124
2. M. G. Shur, “Uniform integrability for strong ratio limit theorems. II”, Theory Probab. Appl., 55:3 (2011), 473–484
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