The Annals of Mathematical Statistics

Strong Ratio Limit Theorems for Markov Processes

Michael Lin

Full-text: Open access


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.

Article information

Ann. Math. Statist., Volume 43, Number 2 (1972), 569-579.

First available in Project Euclid: 27 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier



Lin, Michael. Strong Ratio Limit Theorems for Markov Processes. Ann. Math. Statist. 43 (1972), no. 2, 569--579. doi:10.1214/aoms/1177692637.

Export citation