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

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1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires 2 LMV - Laboratoire de Mathématiques de Versailles

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 ccsd-00002954 . Finally, an application to conditioning on the presence allows to meet again the probability tilting and the so-called additive martingale.

Keywords : probability tilting large deviations time-discretization probability tilting. Fragmentation branching random walk

Author: ** Jean Bertoin - Alain Rouault - **

Source: https://hal.archives-ouvertes.fr/