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April, 1972 Strong Ratio Limit Theorems for Markov Processes
Michael Lin
Ann. Math. Statist. 43(2): 569-579 (April, 1972). DOI: 10.1214/aoms/1177692637


Let $P$ be a conservative and ergodic Markov operator on $L_\infty(X, \Sigma, m)$ (where $m(X) = 1$). It is proved that if for $A \in \Sigma$ with $m(A) > 0$ and $\mu \ll m$ a finite measure with $\mu(A) > 0 \lim_{n\rightarrow\infty} < \mu, P^{n+1}1_B >/<\mu, P^n 1_A>$ exists for every $B \subset A$, then $P$ has a $\sigma$-finite invariant measure $\lambda$ and there is a sequence $A_k \uparrow X$ with $A_0 = A$ such that for $0 \leqq f, g \in L_\infty(A_k) < \mu, P^n f>/<\mu, P^n g>\rightarrow < \lambda, f>/< \lambda, g>$. The result is used to study the convergence of $< \mu, P^n f>/< \eta, P^n g>$ for $\mu, \eta \ll m$, with applications to Harris processes and strong mixing point transformations. An analogous result for a positive contraction of $C(X)$ is given.


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Michael Lin. "Strong Ratio Limit Theorems for Markov Processes." Ann. Math. Statist. 43 (2) 569 - 579, April, 1972.


Published: April, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0243.60039
MathSciNet: MR315787
Digital Object Identifier: 10.1214/aoms/1177692637

Rights: Copyright © 1972 Institute of Mathematical Statistics


Vol.43 • No. 2 • April, 1972
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