Tail probability of maximal displacement in critical and subcritical branching stable processes
Haojie Hou, Yiyang Jiang, Yan-Xia Ren, Renming Song
TL;DR
This work derives exact tail asymptotics for the maximal displacement M in (sub)critical branching α-stable processes (α∈(0,2)). It develops an integral equation for u(x)=P(M≥x) and analyzes it via Laplace transforms, Tauberian theorems, and connections to super α-stable processes to obtain sharp power-law or exponential tails across regimes defined by the offspring mean m and the jump structure (positive vs. spectrally negative). Key results include explicit constants for subcritical positive-jump cases, gamma-domain attraction in the critical regime, and exponential tails in subcritical spectrally negative cases under L log L-type conditions. The methods unify stable-process fluctuation theory with branching mechanisms, extending prior results and providing a versatile framework for extreme-value behavior in branching Lévy systems.
Abstract
In this paper, we study critical and subcritical branching $α$-stable processes, $α\in (0, 2)$. We obtain the exact asymptotic behaviors of the tails of the maximal positions of all subcritical branching $α$-stable processes with positive jumps. In the case of subcritical branching spectrally negative $α$-stable processes, we obtain the exact asymptotic behaviors of the tails of the maximal positions under the assumption that the offspring distributions satisfy the $L\log L$ condition. For critical branching $α$-stable processes, we obtain the exact asymptotic behaviors of the tails under the assumption that the offspring distributions belong to the domain of attraction of a $γ$-distribution, $γ\in (1, 2]$.