## The Annals of Probability

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

Robert Cogburn

#### 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

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