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June, 1957 On Transient Markov Chains with Application to the Uniqueness Problem for Markov Processes
Leo Breiman
Ann. Math. Statist. 28(2): 499-503 (June, 1957). DOI: 10.1214/aoms/1177706979


We focus our attention herein on a Markov chain $x_0, x_1, \cdots$ with a countable number of states indexed by a subset I of the integers and with stationary transition probabilities $p_{ij}$, and explore the sets of states defined by: A transient set of states $C$ is said to be denumerably atomic if $P(x_n \varepsilon C i.o.) > 0$ and if for every infinite set $A \subset C$ we have $x_n \varepsilon C i.o.$ implies $x_n \varepsilon A i.o.$ with probability one (a.s.). Following Blackwell's basic paper [1] which introduced the systematic use of martingales into the study of Markov chains, we use the semi-martingale convergence theorem [2] to characterize denumerably atomic sets in terms of the bounded solutions of the inequality $$\phi(i) \leqq \sum_{j \varepsilon I} p_{ij}\phi(j),\quad i \varepsilon I.$$ For chains whose state space contains a denumerably atomic set a convergence criterion for certain sums $\sum^\infty_{n = 0}f(x_n)$ is then developed. The application of this criterion to a restricted class of continuous parameter Markov processes gives simple necessary and sufficient conditions for the existence of a unique process satisfying given infinitesimal conditions. This last result illuminates the connection between the necessary and sufficient conditions given by Feller [3] for uniqueness and the simpler conditions for birth and death processes given recently by Dobrusin [4], more recently by Karlin and McGregor [5], and by Reuter and Lederman [6] (see also [7]).


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Leo Breiman. "On Transient Markov Chains with Application to the Uniqueness Problem for Markov Processes." Ann. Math. Statist. 28 (2) 499 - 503, June, 1957.


Published: June, 1957
First available in Project Euclid: 27 April 2007

zbMATH: 0078.31703
MathSciNet: MR88101
Digital Object Identifier: 10.1214/aoms/1177706979

Rights: Copyright © 1957 Institute of Mathematical Statistics

Vol.28 • No. 2 • June, 1957
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