The Annals of Probability

A Uniform Theory for Sums of Markov Chain Transition Probabilities

Robert Cogburn

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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


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