The Annals of Probability
- Ann. Probab.
- Volume 33, Number 2 (2005), 645-673.
Criticality for branching processes in random environment
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.
Ann. Probab., Volume 33, Number 2 (2005), 645-673.
First available in Project Euclid: 3 March 2005
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.aop/1109868596