The Annals of Applied Probability

$R$-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata

P. A. Ferrari, H. Kesten, and S. Martínez

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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.

Article information

Ann. Appl. Probab., Volume 6, Number 2 (1996), 577-616.

First available in Project Euclid: 18 October 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Absorbing Markov chain quasi-stationary distribution ratio limit theorem Yaglom limit $R$-positivity central limit theorem


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. 6 (1996), no. 2, 577--616. doi:10.1214/aoap/1034968146.

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