Criticality for branching processes in random environment

Details Criticality for branching processes in random environment Criticality for branching processes in random environment

2005-03-29

arxiv.org/abs/math/0503657v1

Annals of Probability 2005, Vol. 33, No. 2, 645-673

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117904000000928 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117904000000928

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.

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