Necessary and sufficient conditions are given for boundedness of $\sup_n \|\sum^n_{k=1} (P^k(x, \bullet) - P^k(y, \bullet))\|$ and $\sup_n \|\sum^n_{k=1} (P^k(x, \bullet) - \pi\|$, where the norm is total variation and where $\pi$ is an invariant probability measure. Also conditions for convergence of $\sum^\infty_{k=1} (P^k(x, \bullet) - \pi)$ in norm are given. These results require the study of certain "small sets." Two new types are introduced: uniform sets and strongly uniform sets, and these are related to the sets introduced by Harris in his study of the existence of $\sigma$-finite invariant measure.
Publié le : 1975-04-14
Classification:
Markov chain,
general state space,
recurrent in sense of Harris,
sums of transition probabilities,
variational norm,
$D$-set,
uniform set,
strongly uniform set,
regular state,
stability,
compact set,
60J05,
60J10
@article{1176996393,
author = {Cogburn, Robert},
title = {A Uniform Theory for Sums of Markov Chain Transition Probabilities},
journal = {Ann. Probab.},
volume = {3},
number = {6},
year = {1975},
pages = { 191-214},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996393}
}
Cogburn, Robert. A Uniform Theory for Sums of Markov Chain Transition Probabilities. Ann. Probab., Tome 3 (1975) no. 6, pp. 191-214. http://gdmltest.u-ga.fr/item/1176996393/