Asymptotical behaviour of the presence probability in branching random walks and fragmentations

TL;DR

This paper extends the asymptotic analysis of presence probabilities in branching processes and fragmentations, removing previous restrictions on offspring mean and allowing infinite mean scenarios across multiple dimensions.

Contribution

It proves that the asymptotic proportionality of presence probability to expectation holds without the offspring mean condition, even with infinite mean and in higher dimensions.

Findings

Asymptotic presence probability proportional to expectation in supercritical branching random walks.

Extension of results to infinite mean offspring distributions.

Application to homogeneous fragmentations and probability tilting.

Abstract

For a subcritical Galton-Watson process $(\zeta_n)$, it is well known that under an $X \log X$ condition, the quotient $P(\zeta_n > 0)/ E\zeta_n$ has a finite positive limit. There is an analogous result for a (one-dimensional) supercritical branching random walk: when $a$ is in the so-called subcritical speed area, the probability of presence around $na$ in the $n$-th generation is asymptotically proportional to the corresponding expectation. In Rouault (1993) this result was stated under a natural $X \log X$ assumption on the offspring point process and a (unnatural) condition on the offspring mean. Here we prove that the result holds without this latter condition, in particular we allow an infinite mean and a dimension $d \geq 1$ for the state-space. As a consequence the result holds also for homogeneous fragmentations as defined in Bertoin (2001), using the method of discrete-time skeletons; this completes the proof of Theorem 4 in Bertoin-Rouault (2004 see math/PR/0409545). Finally, an application to conditioning on the presence allows to meet again the probability tilting and the so-called additive martingale.