Open Access
November, 1991 A Branching Random Walk with a Barrier
J. D. Biggins, Boris D. Lubachevsky, Adam Shwartz, Alan Weiss
Ann. Appl. Probab. 1(4): 573-581 (November, 1991). DOI: 10.1214/aoap/1177005839

Abstract

Suppose that a child is likely to be weaker than its parent and a child who is too weak will not reproduce. What is the condition for a family line to survive? Let b denote the mean number of children a viable parent will have; we suppose that this is independent of strength as long as strength is positive. Let F denote the distribution of the change in strength from parent to child, and define h=supθ(log(eθtdF(t))). We show that the situation is black or white: 1. If b<eh,thenP(family line dies)=1. 2. If b>eh,thenP(family survives)>0. Define f(x):=E(number of members in the familyinitial strengthx). We show that if b<eh, then there exists a positive constant C such that limxeαxf(x)=C, where α is the smaller of the (at most) two positive roots of bestdF(t)=1. We also find an explicit expression for f(x) when the walk is on a lattice and is skip-free to the left. This process arose in an analysis of rollback-based simulation, and these results are the foundation of that analysis.

Citation

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J. D. Biggins. Boris D. Lubachevsky. Adam Shwartz. Alan Weiss. "A Branching Random Walk with a Barrier." Ann. Appl. Probab. 1 (4) 573 - 581, November, 1991. https://doi.org/10.1214/aoap/1177005839

Information

Published: November, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0749.60076
MathSciNet: MR1129775
Digital Object Identifier: 10.1214/aoap/1177005839

Subjects:
Primary: 60J80
Secondary: 60F10 , 60J15

Keywords: absorbing barrier , branching process , Random walks , survival

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.1 • No. 4 • November, 1991
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