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Persistence of sums of correlated increments and clustering in cellular automata

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  • Lyu, Hanbaek
  • Sivakoff, David

Abstract

Let T be the first return time to (−∞,0] of sums of increments given by a functional of a stationary Markov chain. We determine the asymptotic behavior of the survival probability, P(T≥t)∼Ct−1∕2 for an explicit constant C. Our analysis is based on a 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. As applications, we recover known clustering results for the 3-color cyclic cellular automaton and the Greenberg–Hastings model, and we prove a new clustering result for the 3-color firefly cellular automaton.

Suggested Citation

  • Lyu, Hanbaek & Sivakoff, David, 2019. "Persistence of sums of correlated increments and clustering in cellular automata," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1132-1152.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:4:p:1132-1152
    DOI: 10.1016/j.spa.2018.04.012
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    References listed on IDEAS

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    1. Ivanovs, Jevgenijs, 2017. "Splitting and time reversal for Markov additive processes," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2699-2724.
    2. ARJAS, Elja & SPEED, T.P., 1973. "Symmetric Wiener-Hopf factorisations in Markov additive processes," LIDAM Reprints CORE 134, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Aurzada, Frank & Mukherjee, Sumit, 2023. "Persistence probabilities of weighted sums of stationary Gaussian sequences," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 286-319.
    2. Aurzada, Frank & Buck, Micha & Kilian, Martin, 2020. "Penalizing fractional Brownian motion for being negative," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6625-6637.

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