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December, 1973 Limit Theorems for Reversible Markov Processes
Michael L. Levitan, Lawrence H. Smolowitz
Ann. Probab. 1(6): 1014-1025 (December, 1973). DOI: 10.1214/aop/1176996807


Consider a reversible (self-adjoint) Markov process with a discrete time parameter and stationary transition probability functions satisfying the Harris recurrence condition. $\mathbf{P}^{(n)}(x, S)$ denotes the $n$-step transition probability function from $x$ to the measurable set $S$ and $\pi$ is the sigma-finite stationary measure induced by the above hypotheses. Using both a functional analytic representation for reversible probabilities and probabilistic identities, various limits are considered for both general and discrete spaces. The principle result gives necessary and sufficient conditions for sets $A$ and $\mathbf{B}$ so that a reversible, aperiodic Markov process satisfies the strong ratio limit property $\lim_{m\rightarrow\infty} \mathbf{P}^{(n + k)} (\mu, \mathbf{A})/\mathbf{P}^{(n)} (\nu, \mathbf{B}) = \pi(\mathbf{A})/\pi(\mathbf{B}$ where $\mu$ and v are arbitrary probability distributions defined on the space and $k$ is any integer.


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Michael L. Levitan. Lawrence H. Smolowitz. "Limit Theorems for Reversible Markov Processes." Ann. Probab. 1 (6) 1014 - 1025, December, 1973.


Published: December, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0271.60068
MathSciNet: MR353452
Digital Object Identifier: 10.1214/aop/1176996807

Primary: 60J05

Keywords: Discrete time parameter Markov process , Harris recurrence condition , invariant measure , Markov process with a discrete parameter , ratio limit theorems , reversibility (self-adjointness)

Rights: Copyright © 1973 Institute of Mathematical Statistics


Vol.1 • No. 6 • December, 1973
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