Large deviations and almost sure convergence for the extremes of branching Lévy processes

TL;DR

This paper analyzes the extreme behavior of a supercritical branching Lévy process on $\mathbb{R}$ with regularly varying tails. By combining a one-big-jump lemma, many-to-one reductions, and precise tail asymptotics, it establishes upper and lower large-deviation results for the maximum $R_t$ and for the point process $\mathbb{X}_t$, scaled by $\Lambda(t)$ relative to the heavy-tail scale $h(t)$. It also derives almost-sure convergence results for the maximal displacement and related functionals, showing that $R_t$ ultimately concentrates around the slowly varying scaling $H(e^{-\lambda t}\log t)$ up to a random factor $(\vartheta^*W)^{1/\alpha}$, and characterizes the full limiting structure under conditioning on extremes. The work connects extreme-value limits, random-measure convergence, and almost-sure asymptotics in heavy-tailed branching systems, providing a comprehensive large-deviation and almost-sure framework for extremes of branching Lévy processes.

Abstract

In this paper, we investigate the asymptotic behavior of supercritical branching Markov processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are Lévy processes with regularly varying tails. Recently, Ren et al. [Appl. Probab. 61 (2024)] studied the weak convergence of the extremes of $\{\mathbb{X}_t, t \ge0\}$. In this paper, we establish the large deviation of $\{\mathbb{X}_t, t \ge0\}$ as well as some almost sure convergence results of the maximum of $\mathbb{X}_t$.

Theorems & Definitions (33)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• Lemma 2.2: Many-to-one formula
• Lemma 2.3