## The Annals of Probability

### Strong Ratio Limit Theorems for $\phi$-Recurrent Markov Chains

E. Nummelin

#### Abstract

Let $\{X_n; n = 0, 1, \cdots\}$ be a $\phi$-recurrent Markov chain on a general measurable state space $(S, \mathscr{F})$ with transition probabilities $P(x, A), x \in S, A \in \mathscr{F}$. The convergence of the ratio $\lambda P^{n+m}f / \mu P^ng$ (as $n \rightarrow \infty$), where $\lambda$ and $\mu$ are nonnegative measures on $(S, \mathscr{F})$ and $f$ and $g$ are nonnegative measurable functions on $S$, is studied. We show that the ratio converges, provided that $\lambda, \mu, f$ and $g$ are in a certain sense "small," and provided that for an embedded renewal sequence $\{u(n)\}$ the limit $\lim u(n + 1)/u(n)$ exists.

#### Article information

Source
Ann. Probab., Volume 7, Number 4 (1979), 639-650.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994987

Digital Object Identifier
doi:10.1214/aop/1176994987

Mathematical Reviews number (MathSciNet)
MR537211

Zentralblatt MATH identifier
0411.60069

JSTOR
Nummelin, E. Strong Ratio Limit Theorems for $\phi$-Recurrent Markov Chains. Ann. Probab. 7 (1979), no. 4, 639--650. doi:10.1214/aop/1176994987. https://projecteuclid.org/euclid.aop/1176994987