## The Annals of Probability

- Ann. Probab.
- Volume 3, Number 2 (1975), 191-214.

### A Uniform Theory for Sums of Markov Chain Transition Probabilities

#### Abstract

Necessary and sufficient conditions are given for boundedness of $\sup_n \|\sum^n_{k=1} (P^k(x, \bullet) - P^k(y, \bullet))\|$ and $\sup_n \|\sum^n_{k=1} (P^k(x, \bullet) - \pi\|$, where the norm is total variation and where $\pi$ is an invariant probability measure. Also conditions for convergence of $\sum^\infty_{k=1} (P^k(x, \bullet) - \pi)$ in norm are given. These results require the study of certain "small sets." Two new types are introduced: uniform sets and strongly uniform sets, and these are related to the sets introduced by Harris in his study of the existence of $\sigma$-finite invariant measure.

#### Article information

**Source**

Ann. Probab. Volume 3, Number 2 (1975), 191-214.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996393

**Digital Object Identifier**

doi:10.1214/aop/1176996393

**Mathematical Reviews number (MathSciNet)**

MR378103

**Zentralblatt MATH identifier**

0348.60106

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J05: Discrete-time Markov processes on general state spaces

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

**Keywords**

Markov chain general state space recurrent in sense of Harris sums of transition probabilities variational norm $D$-set uniform set strongly uniform set regular state stability compact set

#### Citation

Cogburn, Robert. A Uniform Theory for Sums of Markov Chain Transition Probabilities. Ann. Probab. 3 (1975), no. 2, 191--214. doi:10.1214/aop/1176996393. https://projecteuclid.org/euclid.aop/1176996393