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Path to survival for the critical branching processes in a random environment

Part of: Stochastic processes Limit theorems Special processes Markov processes

Published online by Cambridge University Press:  22 June 2017

Vladimir Vatutin  and
Elena Dyakonova
Vladimir Vatutin*
Affiliation:
Steklov Mathematical Institute
Elena Dyakonova*
Affiliation:
Steklov Mathematical Institute
*
* Postal address: Steklov Mathematical Institute, Gubkin Street 8, Moscow 119991, Russia.
* Postal address: Steklov Mathematical Institute, Gubkin Street 8, Moscow 119991, Russia.
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Abstract

A critical branching process {Zk, k = 0, 1, 2, ...} in a random environment is considered. A conditional functional limit theorem for the properly scaled process {log Zpu, 0 ≤ u < ∞} is established under the assumptions that Zn > 0 and p ≪ n. It is shown that the limiting process is a Lévy process conditioned to stay nonnegative. The proof of this result is based on a limit theorem describing the distribution of the initial part of the trajectories of a driftless random walk conditioned to stay nonnegative.

Keywords

Branching processrandom environmentrandom walk conditioned to stay positiveLévy process conditioned to stay positivechange of measurefunctional limit theorem

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K37: Processes in random environments 60G50: Sums of independent random variables; random walks 60F17: Functional limit theorems; invariance principles
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 588 - 602
Copyright
Copyright © Applied Probability Trust 2017 

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