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A second order asymptotic expansion in the local limit theorem for a simple branching random walk in Zd

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  • Gao, Zhi-Qiang

Abstract

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton–Watson process and the migration of particles by a simple random walk in Zd. Denote by Zn(z) the number of particles of generation n located at site z∈Zd. We give the second order asymptotic expansion for Zn(z). The higher order expansion can be derived by using our method here. As a by-product, we give the second order expansion for a simple random walk on Zd, which is used in the proof of the main theorem and is of independent interest.

Suggested Citation

  • Gao, Zhi-Qiang, 2018. "A second order asymptotic expansion in the local limit theorem for a simple branching random walk in Zd," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4000-4017.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:12:p:4000-4017
    DOI: 10.1016/j.spa.2018.01.005
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    References listed on IDEAS

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    1. Asmussen, Soren & Kaplan, Norman, 1976. "Branching random walks I," Stochastic Processes and their Applications, Elsevier, vol. 4(1), pages 1-13, January.
    2. 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.
    3. 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.
    4. 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.
    5. Kaplan, Norman & Asmussen, Soren, 1976. "Branching random walks II," Stochastic Processes and their Applications, Elsevier, vol. 4(1), pages 15-31, January.
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