We prove that certain (discrete time) probabilistic automata which
can be absorbed in a "null state" have a normalized quasi-stationary
distribution (when restricted to the states other than the null state). We also
show that the conditional distribution of these systems, given that they are
not absorbed before time n, converges to an honest probability
distribution; this limit distribution is concentrated on the configurations
with only finitely many "active or occupied" sites.
¶ A simple example to which our results apply is the discrete time
version of the subcritical contact process on $\mathbb{Z}^d$ or oriented
percolation on $\mathbb{Z}^d$ (for any $d \geq 1$) as seen from the "leftmost
particle." For this and some related models we prove in addition a central
limit theorem for $n^{-1/2}$ times the position of the leftmost particle
(conditioned on survival until time n).
¶ The basic tool is to prove
that our systems are R-positive-recurrent.
Publié le : 1996-05-14
Classification:
Absorbing Markov chain,
quasi-stationary distribution,
ratio limit theorem,
Yaglom limit,
$R$-positivity,
central limit theorem,
60J10,
60F05,
60K35
@article{1034968146,
author = {Ferrari, P. A. and Kesten, H. and Mart\'\i nez, S.},
title = {$R$-positivity, quasi-stationary distributions and ratio limit
theorems for a class of probabilistic automata},
journal = {Ann. Appl. Probab.},
volume = {6},
number = {1},
year = {1996},
pages = { 577-616},
language = {en},
url = {http://dml.mathdoc.fr/item/1034968146}
}
Ferrari, P. A.; Kesten, H.; Martínez, S. $R$-positivity, quasi-stationary distributions and ratio limit
theorems for a class of probabilistic automata. Ann. Appl. Probab., Tome 6 (1996) no. 1, pp. 577-616. http://gdmltest.u-ga.fr/item/1034968146/