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

Abstract

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.