## The Annals of Mathematical Statistics

### Strong Ratio Limit Theorems for Markov Processes

Michael Lin

#### Abstract

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

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

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177692637

Digital Object Identifier
doi:10.1214/aoms/1177692637

Mathematical Reviews number (MathSciNet)
MR315787

Zentralblatt MATH identifier
0243.60039

JSTOR