The Annals of Probability

Criticality for branching processes in random environment

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

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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.

Article information

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

First available in Project Euclid: 3 March 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G50: Sums of independent random variables; random walks 60F17: Functional limit theorems; invariance principles

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


Afanasyev, V. I.; Geiger, J.; Kersting, G.; Vatutin, V. A. Criticality for branching processes in random environment. Ann. Probab. 33 (2005), no. 2, 645--673. doi:10.1214/009117904000000928.

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