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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 $h = \sup_\theta(-\log(\int e^{\theta t} dF(t)))$. We show that the situation is black or white: 1. If $b < e^h, \text{then} P(\text{family line dies}) = 1$. 2. If $b > e^h, \text{then} P(\text{family survives}) > 0$. Define $f(x) := E(\text{number of members in the family} \mid \text{initial strength} x)$. We show that if $b < e^h$, then there exists a positive constant $C$ such that $\lim_{x \rightarrow \infty}e^{- \alpha x}f(x) = C$, where $\alpha$ is the smaller of the (at most) two positive roots of $b \int e^{st} dF(t) = 1$. 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.

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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

Rights: Copyright © 1991 Institute of Mathematical Statistics

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Vol.1 • No. 4 • November, 1991
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