Tail probability of maximal displacement in critical branching L\'{e}vy process with stable branching

TL;DR

This paper investigates the tail probability of the maximal displacement in a critical branching Lévy process with stable offspring distribution, generalizing previous results for branching Brownian motion and random walks.

Contribution

It extends existing results by analyzing the tail behavior of maximal displacement in a critical branching Lévy process with stable offspring distribution, relaxing the finite third moment condition.

Findings

Derived the tail probability asymptotics for the maximal displacement.

Generalized classical results to Lévy processes with stable offspring distribution.

Established conditions under which the tail behavior follows a specific asymptotic form.

Abstract

Consider a critical branching L\'{e}vy process $\{X_t, t\ge 0\}$ with branching rate $\beta>0, $ offspring distribution $\{p_k:k\geq 0\}$ and spatial motion $\{\xi_t, \Pi_x\}$. For any $t\ge 0$, let $N_t$ be the collection of particles alive at time $t$, and, for any $u\in N_t$, let $X_u(t)$ be the position of $u$ at time $t$. We study the tail probability of the maximal displacement $M:=\sup_{t>0}\sup_{u\in N_t} X_u(t)$ under the assumption $\lim_{n\to\infty} n^\alpha \sum_{k=n}^\infty p_k =\kappa\in(0,\infty)$ for some $\alpha\in (1,2)$, $\Pi_0(\xi_1)=0$ and $\Pi_0 (|\xi_1|^r)\in (0,\infty)$ for some $r> 2\alpha/(\alpha-1)$. Our main result is a generalization of the main result of Sawyer and Fleischman (1979) for branching Brownian motions and that of Lalley and Shao (2015) for branching random walks, both of which are proved under the assumption $\sum_{k=0}^\infty k^3 p_k<\infty$.