## The Annals of Probability

- Ann. Probab.
- Volume 10, Number 4 (1982), 942-954.

### Renewal Theory for Markov Chains on the Real Line

#### Abstract

Standard renewal theory is concerned with expectations related to sums of positive i.i.d. variables, $S_n = \sum^n_{i=1} Z_i$. We generalize this theory to the case where $\{S_i\}$ is a Markov chain on the real line with stationary transition probabilities satisfying a drift condition. The expectations we are concerned with satisfy generalized renewal equations, and in our main theorems, we show that these expectations are the unique solutions of the equations they satisfy.

#### Article information

**Source**

Ann. Probab., Volume 10, Number 4 (1982), 942-954.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993716

**Digital Object Identifier**

doi:10.1214/aop/1176993716

**Mathematical Reviews number (MathSciNet)**

MR672295

**Zentralblatt MATH identifier**

0498.60087

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62L05: Sequential design

Secondary: 62K20: Response surface designs

**Keywords**

Renewal theory Markov chains random walks

#### Citation

Keener, Robert W. Renewal Theory for Markov Chains on the Real Line. Ann. Probab. 10 (1982), no. 4, 942--954. doi:10.1214/aop/1176993716. https://projecteuclid.org/euclid.aop/1176993716