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Exact convergence rate in the local central limit theorem for a lattice branching random walk on Zd

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  • Gao, Zhi-Qiang

Abstract

Consider a branching random walk, where the branching mechanism is governed by a Galton–Watson process and the migration is governed by a lattice random walk on Zd. Under the mild moment conditions for the underlying branching mechanism and migration laws, we figure out the exact convergence rate of the local limit theorem for the counting measure which counts the number of particles of generation n in a given set. Our result extends the previous one obtained by Chen (2001) for a simple branching random walk with second moment condition for the offspring law.

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  • Gao, Zhi-Qiang, 2019. "Exact convergence rate in the local central limit theorem for a lattice branching random walk on Zd," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 58-66.
    • Handle: RePEc:eee:stapro:v:151:y:2019:i:c:p:58-66
    DOI: 10.1016/j.spl.2019.03.016
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    References listed on IDEAS

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    1. Gao, Zhiqiang & Liu, Quansheng, 2016. "Exact convergence rates in central limit theorems for a branching random walk with a random environment in time," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2634-2664.
    2. Kaplan, Norman & Asmussen, Soren, 1976. "Branching random walks II," Stochastic Processes and their Applications, Elsevier, vol. 4(1), pages 15-31, January.
    3. Asmussen, Soren & Kaplan, Norman, 1976. "Branching random walks I," Stochastic Processes and their Applications, Elsevier, vol. 4(1), pages 1-13, January.
    4. Biggins, J. D., 1990. "The central limit theorem for the supercritical branching random walk, and related results," Stochastic Processes and their Applications, Elsevier, vol. 34(2), pages 255-274, April.
    5. Gao, Zhiqiang, 2017. "Exact convergence rate of the local limit theorem for branching random walks on the integer lattice," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1282-1296.
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