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

Tweedie, Richard L.

Ann. Probab., Tome 2 (1974) no. 6, p. 865-878 / Harvested from Project Euclid

Résumé

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 $\mathscr{X}$.

Publié le : 1974-10-14
Classification: $R$-theory, Markov chains, invariant measures, subinvariant measures, potential theory, boundary theory, stationary measures, integral equations, 60J05, 60J45, 60J50, 45C05, 45N05

@article{1176996553,
     author = {Tweedie, Richard L.},
     title = {$R$-Theory for Markov Chains on a General State Space II: $r$-Subinvariant Measures for $r$-Transient Chains},
     journal = {Ann. Probab.},
     volume = {2},
     number = {6},
     year = {1974},
     pages = { 865-878},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176996553}
}

Tweedie, Richard L. $R$-Theory for Markov Chains on a General State Space II: $r$-Subinvariant Measures for $r$-Transient Chains. Ann. Probab., Tome 2 (1974) no. 6, pp.  865-878. http://gdmltest.u-ga.fr/item/1176996553/