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

Full-text: Open access

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 $\mathscr{X}$.

Article information

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

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996553

Digital Object Identifier
doi:10.1214/aop/1176996553

Mathematical Reviews number (MathSciNet)
MR368152

Zentralblatt MATH identifier
0296.60041

JSTOR
links.jstor.org

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aop/1176996553


Export citation

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.