Branching-stable point measures and processes

TL;DR

This paper introduces branching-stable point measures, exploring their properties, asymptotic behavior, and genealogical structure, revealing a rich family of processes with negative scaling exponents related to Crump-Mode-Jagers branching processes.

Contribution

It defines and analyzes branching-stable point measures, extending stable process theory to branching mechanisms and characterizing their asymptotics and genealogies.

Findings

Existence of a rich family of branching-stable point measures with negative scaling exponents.

Asymptotic behavior of the number of atoms in $(- abla, x]$ as $x o abla$.

Genealogical structures of typical atoms are characterized.

Abstract

We introduce and study the class of branching-stable point measures, which can be seen as an analog of stable random variables when the branching mechanism for point measures replaces the usual addition. In contrast with the classical theory of stable (L\'evy) processes, there exists a rich family of branching-stable point measures with \emph{negative} scaling exponent, which can be described as certain Crump-Mode-Jagers branching processes. We investigate the asymptotic behavior of their cumulative distribution functions, that is, the number of atoms in $(-\infty, x]$ as $x\to \infty$, and further depict the genealogical lineage of typical atoms. For both results, we rely crucially on the work of Biggins.