On the maximal displacement of some critical branching L{é}vy processes with stable offspring distribution

TL;DR

The paper analyzes the maximal displacement $M$ of a critical branching Lévy process where the offspring distribution lies in the domain of attraction of a stable law of index $\beta\in(1,2)$. It develops an integral equation for $u(x)=\mathbb{P}(M\ge x)$ and solves it via Fourier/Wiener-Hopf methods and Tauberian arguments, obtaining precise tail asymptotics in two regimes: (i) finite second moments with nonzero drift for the Lévy process $L$, and (ii) regularly varying tails of $L$ with index $\alpha\in(0,2)$. In regime (i), if $\mathbb{E}[L_1]>0$, $\mathbb{P}(M\ge x)$ decays as $x^{-1/(\beta-1)}$ with a constant depending on $\mathbb{E}[L_1]$, while if $\mathbb{E}[L_1]<0$, it decays exponentially with rate $\omega$ solving $\Psi(\omega)=0$. In regime (ii), the tail of $M$ is heavy with $\mathbb{P}(M\ge x) \sim \ell_\alpha^{1/\beta}(x) x^{-\alpha/\beta}$ up to a constant, covering the cases $\alpha\in(0,1]$, $\alpha\in(1,2)$, and the boundary $\alpha=1$. These results extend prior work by allowing non-centered Lévy motions and offspring in the $\beta$-stable domain, and by treating heavy-tailed Lévy drivers without imposing strong moment conditions on the offspring distribution.

Abstract

Let X be a critical branching L{é}vy process whose offspring distribution is in the domain of attraction of a stable random variable. We study the tail probability of the maximum location ever reached by a particle in two different situations: first when the underlying L{é}vy process L admits moments of order at least two and is not centered, and then when the distribution of L has a regularly varying tail. This work complements some earlier results in which either L was centered or the offspring distribution was assumed to have moments of order three.

Theorems & Definitions (20)

• Theorem 1
• Theorem 3
• Remark 4
• Lemma 5
• proof
• Lemma 6
• proof
• Remark 7
• Lemma 8
• proof