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Survival probabilities and maxima of sums of correlated increments with applications to one-dimensional cellular automata. (arXiv:1706.08117v1 [math.PR])
来源于:arXiv
We consider sums of increments given by a functional of a stationary Markov
chain. Letting $T$ be the first return time of the partial sums process to
$(-\infty,0]$, under general assumptions we give determine the asymptotic
behavior of the survival probability, $\mathbb{P}(T\ge t)\sim Ct^{-1/2}$ for an
explicit constant $C$. Our analysis is based on a novel connection between the
survival probability and the running maximum of the time-reversed process, and
relies on a functional central limit theorem for Markov chains. Our result
extends the classic theorem of Sparre Anderson on sums of mean zero and
independent increments to the case of correlated increments. As applications,
we recover known clustering results for the 3-color cyclic cellular automaton
and the Greenberg-Hastings model in one dimension, and we prove a new
clustering result for the 3-color firefly cellular automaton. 查看全文>>