2004/09/28 by Jean Bertoin, Bertoin, Jean, Alain Rouault +1 · 1 citation
Environmental Science · Mathematics · Physics and Astronomy · #Coagulation and Flocculation Studies #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:10 #msc:25 #msc:60
paper · pdf · doi:10.48550/arxiv.math/0409547
15 pages, companion paper of math.PR/0409545
arxiv created 2004/09/28 · arxiv updated 2009/12/01
For a subcritical Galton-Watson process (ζn), it is well known that under an X log X condition, the quotient P(ζn > 0)/ Eζn 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 ≥ 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.