The Annals of Probability

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

Richard L. Tweedie

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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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Ann. Probab., Volume 2, Number 5 (1974), 865-878.

First available in Project Euclid: 19 April 2007

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Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J50: Boundary theory 45C05: Eigenvalue problems [See also 34Lxx, 35Pxx, 45P05, 47A75] 45N05: Abstract integral equations, integral equations in abstract spaces

$R$-theory Markov chains invariant measures subinvariant measures potential theory boundary theory stationary measures integral equations


Tweedie, Richard L. $R$-Theory for Markov Chains on a General State Space II: $r$-Subinvariant Measures for $r$-Transient Chains. Ann. Probab. 2 (1974), no. 5, 865--878. doi:10.1214/aop/1176996553.

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See also

  • Part I: Richard L. Tweedie. $R$-Theory for Markov Chains on a General State Space I: Solidarity Properties and $R$-Recurrent Chains. Ann. Probab., Volume 2, Number 5 (1974), 840--864.