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March 2005 Criticality for branching processes in random environment
V. I. Afanasyev, J. Geiger, G. Kersting, V. A. Vatutin
Ann. Probab. 33(2): 645-673 (March 2005). DOI: 10.1214/009117904000000928

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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V. I. Afanasyev. J. Geiger. G. Kersting. V. A. Vatutin. "Criticality for branching processes in random environment." Ann. Probab. 33 (2) 645 - 673, March 2005. https://doi.org/10.1214/009117904000000928

Information

Published: March 2005
First available in Project Euclid: 3 March 2005

zbMATH: 1075.60107
MathSciNet: MR2123206
Digital Object Identifier: 10.1214/009117904000000928

Subjects:
Primary: 60J80
Secondary: 60F17 , 60G50

Keywords: branching process , Conditioned random walk , Functional limit theorem , random environment , Random walk , Spitzer’s condition , Tanaka decomposition

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 2 • March 2005
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