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Reduced subcritical Galton-Watson processes in a random environment

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Klaus Fleischmann  and
Vladimir A. Vatutin
Klaus Fleischmann*
Affiliation:
Weierstrass Institute, Berlin
Vladimir A. Vatutin*
Affiliation:
Steklov Mathematical Institute, Moscow
*
∗ Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany. Email address: fleischmann@wias-berlin.de
∗∗ Postal address: Department of Discrete Mathematics, Steklov Mathematical Institute, 8 Gubkin Street, 117 966 Moscow, GSP-1, Russia.
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Abstract

We study the structure of genealogical trees of reduced subcritical Galton-Watson processes in a random environment assuming that all (randomly varying in time) offspring generating functions are fractional linear. We show that this structure may differ significantly from that of the ‘classical’ reduced subcritical Galton-Watson processes. In particular, it may look like a complex ‘hybrid’ of classical reduced super and subcritical processes. Some relations with random walks in a random environment are discussed.

Keywords

Random environmentbranching processrandom walkreduced processreduced treeconditional limit theoremsource timehybrid behaviorladder epochs

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 88 - 111
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Supported in part by the Grant RFBR N 96-00338, INTAS-RFBR 95-0099, and the DFG.

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