Stable-like fluctuations of Biggins’ martingales

Alexander Iksanov, Konrad Kolesko and Matthias Meiners

Stochastic Processes and their Applications, 2019, vol. 129, issue 11, 4480-4499

Abstract: Let (Wn(θ))n∈N0 be Biggins’ martingale associated with a supercritical branching random walk, and let W(θ) be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of W1(θ) belongs to the domain of normal attraction of an α-stable distribution for some α∈(1,2), then, as n→∞, there is weak convergence of the tail process (W(θ)−Wn−k(θ))k∈N0, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with α-stable marginals.

Keywords: Autoregressive process; Biggins’ martingale; Branching random walk; Martingale; Stable distribution (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:129:y:2019:i:11:p:4480-4499

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DOI: 10.1016/j.spa.2018.11.022

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