$R$-Theory for Markov Chains on a General State Space II: $r$-Subinvariant Measures for $r$-Transient Chains

  • Tweedie R
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Abstract

This paper is a sequel to a previous paper of similar title. The structure of $r$-subinvariant measures for a Markov chain $\{X_n\}$ on a general state space $(\mathscr{X}, \mathscr{F})$ is investigated in the $r$-transient case, and a Martin boundary representation is found. Under certain continuity assumptions on the transition law of $\{X_n\}$ the elements of the Martin boundary are identified when $\mathscr{F}$ is countably generated, and a necessary and sufficient condition for an $r$-invariant measure for $\{X_n\}$ to exist is found. This generalizes the Harris-Veech conditions for countable $\mathscr{X}$.

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Tweedie, R. L. (2007). $R$-Theory for Markov Chains on a General State Space II: $r$-Subinvariant Measures for $r$-Transient Chains. The Annals of Probability, 2(5). https://doi.org/10.1214/aop/1176996553

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