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April, 1975 A Uniform Theory for Sums of Markov Chain Transition Probabilities
Robert Cogburn
Ann. Probab. 3(2): 191-214 (April, 1975). DOI: 10.1214/aop/1176996393

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.

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Robert Cogburn. "A Uniform Theory for Sums of Markov Chain Transition Probabilities." Ann. Probab. 3 (2) 191 - 214, April, 1975. https://doi.org/10.1214/aop/1176996393

Information

Published: April, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0348.60106
MathSciNet: MR378103
Digital Object Identifier: 10.1214/aop/1176996393

Subjects:
Primary: 60J05
Secondary: 60J10

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

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 2 • April, 1975
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