Asymptotical behaviour of the presence probability in branching random walks and fragmentations
| dc.creator | Bertoin, Jean | |
| dc.creator | Rouault, Alain | |
| dc.date | 2004-09-28 | |
| dc.date.accessioned | 2026-07-07T05:12:40Z | |
| dc.date.available | 2026-07-07T05:12:40Z | |
| dc.description | For a subcritical Galton-Watson process $(ζ_n)$, it is well known that under an $X \log X$ condition, the quotient $P(ζ_n > 0)/ Eζ_n$ 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. | |
| dc.description | 15 pages, companion paper of math.PR/0409545 | |
| dc.identifier | https://arxiv.org/abs/math/0409547 | |
| dc.identifier | http://arxiv.org/abs/math/0409547 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72658 | |
| dc.subject | Probability | |
| dc.subject | 60 J 25, 60 F 10 | |
| dc.title | Asymptotical behaviour of the presence probability in branching random walks and fragmentations | |
| dc.type | text |