Asymptotical behaviour of the presence probability in branching random walks and fragmentations - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2004

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

Alain Rouault

Résumé

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.
Fichier principal
Vignette du fichier
presence.pdf (315.52 Ko) Télécharger le fichier

Dates et versions

hal-00002955 , version 1 (28-09-2004)

Identifiants

  • HAL Id : hal-00002955 , version 1

Citer

Jean Bertoin, Alain Rouault. Asymptotical behaviour of the presence probability in branching random walks and fragmentations. 2004. ⟨hal-00002955⟩
271 Consultations
132 Téléchargements

Partager

Gmail Facebook X LinkedIn More