Criticality for branching processes in random environment

Criticality for branching processes in random environment

V. I. Afanasyev, J. Geiger, G. Kersting, and V. A. Vatutin

Source: Ann. Probab. Volume 33, Number 2 (2005), 645-673.

Abstract

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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Primary Subjects: 60J80

Secondary Subjects: 60G50, 60F17

Keywords: Branching process; random environment; random walk; conditioned random walk; Spitzer’s condition; Tanaka decomposition; functional limit theorem

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1109868596
Digital Object Identifier: doi:10.1214/009117904000000928
Mathematical Reviews number (MathSciNet): MR2123206
Zentralblatt MATH identifier: 02164478

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