Maximal Displacement of Branching Symmetric Stable Processes

Shiozawa, Yuichi

doi:10.1007/978-981-19-4672-1_23

Cited by 2 publications

(2 citation statements)

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“…In this paper, we study the extremes of branching Lévy processes with constant branching rate β (that is, μ(dx) = β dx) and spatial motion having regularly varying tails (see condition (H2)). Since β dx is not compactly supported, we cannot get the growth rate of the maximal displacement from [38]. As a corollary of our extreme limit result we get the growth rate of the maximal displacement, see Corollary 1.2.…”

Section: Introductionmentioning

confidence: 86%

“…the support of μ is compact) and the offspring distribution {p n (x), n ≥ 0} being spatially dependent. Under some conditions on μ and {p n (x), n ≥ 0}, [38] proved that the growth rate of the maximal displacement is exponential with rate given by the principal eigenvalue of the mean semigroup of the branching symmetric stable process. In this paper, we study the extremes of branching Lévy processes with constant branching rate β (that is, μ(dx) = β dx) and spatial motion having regularly varying tails (see condition (H2)).…”

Section: Introductionmentioning

confidence: 99%

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Weak convergence of the extremes of branching Lévy processes with regularly varying tails

Ren,

Song,

Zhang

2023

J. Appl. Probab.

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We study the weak convergence of the extremes of supercritical branching Lévy processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are Lévy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, $\mathbb{X}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of $\mathbb{X}_t$ .

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Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%