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The central limit theorem for the supercritical branching random walk, and related results

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  • Biggins, J. D.

Abstract

In a branching random walk each family arrives equipped with its members' positions relative to their parent, these being i.i.d. copies of some point process X. The supercritical case is considered so the mean family size m> 1, and X has intensity measure m[nu], where [nu] is a probability measure. The nth generation of the process, Z(n), then has intensity measure mn[nu]n* so it is natural to expect m-nZ(n) to exhibit 'central limit' behaviour. Such a result, corresponding results for stable laws, their local analogues and some similar results when a Seneta-Heyde norming is needed are all obtained here. The main result, from which these are derived, provides a good approximation to the 'characteristic function' of Z(n) for the generation dependent version of the process.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:34:y:1990:i:2:p:255-274
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    Citations

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    Cited by:

    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. Bertoin, Jean, 2006. "Different aspects of a random fragmentation model," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 345-369, March.
    3. Huang, Chunmao & Liu, Quansheng, 2024. "Limit theorems for a branching random walk in a random or varying environment," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    4. 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.
    5. Xiaoqiang Wang & Chunmao Huang, 2017. "Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time," Journal of Theoretical Probability, Springer, vol. 30(3), pages 961-995, September.
    6. Shi, Wanlin, 2019. "A note on large deviation probabilities for empirical distribution of branching random walks," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 18-28.
    7. Vincent Bansaye, 2019. "Ancestral Lineages and Limit Theorems for Branching Markov Chains in Varying Environment," Journal of Theoretical Probability, Springer, vol. 32(1), pages 249-281, March.
    8. Johnson, Torrey, 2014. "On the support of the simple branching random walk," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 107-109.
    9. Haojie Hou & Yan-Xia Ren & Renming Song, 2024. "Asymptotic Expansions for Additive Measures of Branching Brownian Motions," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3355-3394, November.
    10. 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.
    11. Biggins, J. D. & Cohn, H. & Nerman, O., 1999. "Multi-type branching in varying environment," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 357-400, October.
    12. 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.

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