We study branching processes in an i.i.d. random environment, where the associated random walk is of the oscillating type. This class of processes generalizes the classical notion of criticality. The main properties of such branching processes are developed under a general assumption, known as Spitzer’s condition in fluctuation theory of random walks, and some additional moment condition. We determine the exact asymptotic behavior of the survival probability and prove conditional functional limit theorems for the generation size process and the associated random walk. The results rely on a stimulating interplay between branching process theory and fluctuation theory of random walks.
Publié le : 2005-03-14
Classification:
Branching process,
random environment,
random walk,
conditioned random walk,
Spitzer’s condition,
Tanaka decomposition,
functional limit theorem,
60J80,
60G50,
60F17
@article{1109868596,
author = {Afanasyev, V. I. and Geiger, J. and Kersting, G. and Vatutin, V. A.},
title = {Criticality for branching processes in random environment},
journal = {Ann. Probab.},
volume = {33},
number = {1},
year = {2005},
pages = { 645-673},
language = {en},
url = {http://dml.mathdoc.fr/item/1109868596}
}
Afanasyev, V. I.; Geiger, J.; Kersting, G.; Vatutin, V. A. Criticality for branching processes in random environment. Ann. Probab., Tome 33 (2005) no. 1, pp. 645-673. http://gdmltest.u-ga.fr/item/1109868596/