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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. Received: 23.03.2004
Revised: 15.02.2005
DOI:
10.4213/tvp92
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