On the law of terminal value of additive martingales in a remarkable branching stable process

TL;DR

This paper provides an explicit description of the law of the terminal value of additive martingales in a special branching stable process, revealing unique tail behaviors and the self-decomposable nature of its distribution.

Contribution

It offers a novel explicit characterization of the terminal value's distribution, including tail decay rates and self-decomposability, contrasting with previous literature.

Findings

Right tail probability decays exponentially

Left tail probability follows a quadratic logarithmic decay

Distribution is self-decomposable and unimodal

Abstract

We give an explicit description of the law of terminal value $W$ of additive martingales in a remarkable branching stable process. We show that the right tail probability of the terminal value decays exponentially fast and the left tail probability follows that $-\log \mathbb{P}(W<x) \sim \frac{1}{2} (\log x)^2$ as $x \rightarrow 0+$. These are in sharp contrast with results in the literature such as Liu (2000, 2001) and Buraczewski (2009). We further show that the law of $W$ is self-decomposable, and therefore, possesses a unimodal density. We specify the asymptotic behavior at $0$ and at $+\infty$ of the latter.