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.
Publié le : 2004-09-28
Classification:
probability tilting,
large deviations,
time-discretization,
probability tilting.,
Fragmentation,
branching random walk,
60 J 25, 60 F 10, 60J80,
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
@article{hal-00002955,
author = {Bertoin, Jean and Rouault, Alain},
title = {Asymptotical behaviour of the presence probability in branching random walks and fragmentations},
journal = {HAL},
volume = {2004},
number = {0},
year = {2004},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00002955}
}
Bertoin, Jean; Rouault, Alain. Asymptotical behaviour of the presence probability in branching random walks and fragmentations. HAL, Tome 2004 (2004) no. 0, . http://gdmltest.u-ga.fr/item/hal-00002955/