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Large deviation probabilities for the range of a d-dimensional supercritical branching random walk

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  • Zhang, Shuxiong

Abstract

Let {Zn}n≥0 be a d-dimensional supercritical branching random walk started from the origin. Write Zn(S) for the number of particles located in a set S⊂Rd at time n. Denote by Rn:=inf⁡{ρ≥0:Zi({|x|≥ρ})=0,∀0≤i≤n} the radius of the minimal ball (centered at the origin) containing the range of {Zi}i≥0 up to time n. In this work, we show that under some mild conditions Rn/n converges in probability to some positive constant x⁎ as n→∞. Furthermore, we study its corresponding lower and upper deviation probabilities, i.e. the decay rates ofP(Rn≤xn)forx∈(0,x⁎);P(Rn≥xn)forx∈(x⁎,∞) as n→∞.

Suggested Citation

  • Zhang, Shuxiong, 2024. "Large deviation probabilities for the range of a d-dimensional supercritical branching random walk," Applied Mathematics and Computation, Elsevier, vol. 462(C).
    • Handle: RePEc:eee:apmaco:v:462:y:2024:i:c:s0096300323005131
    DOI: 10.1016/j.amc.2023.128344
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    References listed on IDEAS

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    1. Shuxiong Zhang, 2023. "Lower Deviation Probabilities for Level Sets of the Branching Random Walk," Journal of Theoretical Probability, Springer, vol. 36(2), pages 811-844, June.
    2. Ren, Yan-Xia & Song, Renming & Zhang, Rui, 2021. "The extremal process of super-Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 137(C), pages 1-34.
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