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Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time

Author

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  • Xiaoqiang Wang

    (Shandong University (Weihai))

  • Chunmao Huang

    (Harbin Institute of Technology at Weihai)

Abstract

We consider a branching random walk on $${\mathbb {R}}$$ R with a stationary and ergodic environment $$\xi =(\xi _n)$$ ξ = ( ξ n ) indexed by time $$n\in {\mathbb {N}}$$ n ∈ N . Let $$Z_n$$ Z n be the counting measure of particles of generation n and $$\tilde{Z}_n(t)=\int \mathrm{e}^{tx}Z_n(\mathrm{d}x)$$ Z ~ n ( t ) = ∫ e t x Z n ( d x ) be its Laplace transform. We show the $$L^p$$ L p convergence rate and the uniform convergence of the martingale $$\tilde{Z}_n(t)/{\mathbb {E}}[\tilde{Z}_n(t)|\xi ]$$ Z ~ n ( t ) / E [ Z ~ n ( t ) | ξ ] , and establish a moderate deviation principle for the measures $$Z_n$$ Z n .

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:3:d:10.1007_s10959-016-0668-6
    DOI: 10.1007/s10959-016-0668-6
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    References listed on IDEAS

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    1. Francis Comets & Nobuo Yoshida, 2011. "Branching Random Walks in Space–Time Random Environment: Survival Probability, Global and Local Growth Rates," Journal of Theoretical Probability, Springer, vol. 24(3), pages 657-687, September.
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    Cited by:

    1. 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).

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