Persistence of sums of correlated increments and clustering in cellular\n automata

2017/06/25 by Hanbaek Lyu, Lyu, Hanbaek, David Sivakoff +1 · 1 citation
Computer Science · Mathematics · #Cellular Automata and Applications #Cellular Automata and Lattice Gases (nlin.CG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1706.08117

openalex publication_date 2017/06/25 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

We consider sums of increments given by a functional of a stationary Markov\nchain. Letting T be the first return time of the partial sums process to\n(-\∞,0], under general assumptions, we determine the asymptotic behavior\nof the survival probability, \ℙ(T\≥ t)\∼ Ct^-1/2 for an explicit\nconstant C. Our analysis is based on a connection between the survival\nprobability and the running maximum of the time-reversed process, and relies on\na functional central limit theorem for Markov chains. Our result extends the\nclassic theorem of Sparre Anderson on sums of mean zero and independent\nincrements to the case of correlated increments. As applications, we recover\nknown clustering results for the 3-color cyclic cellular automaton and the\nGreenberg-Hastings model in one dimension, and we prove a new clustering result\nfor the 3-color firefly cellular automaton.\n

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