The Annals of Applied Probability

A Branching Random Walk with a Barrier

J. D. Biggins, Boris D. Lubachevsky, Adam Shwartz, and Alan Weiss

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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.

Article information

Source
Ann. Appl. Probab., Volume 1, Number 4 (1991), 573-581.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177005839

Digital Object Identifier
doi:10.1214/aoap/1177005839

Mathematical Reviews number (MathSciNet)
MR1129775

Zentralblatt MATH identifier
0749.60076

JSTOR
links.jstor.org

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J15 60F10: Large deviations

Keywords
Branching process random walks absorbing barrier survival

Citation

Biggins, J. D.; Lubachevsky, Boris D.; Shwartz, Adam; Weiss, Alan. A Branching Random Walk with a Barrier. Ann. Appl. Probab. 1 (1991), no. 4, 573--581. doi:10.1214/aoap/1177005839. https://projecteuclid.org/euclid.aoap/1177005839


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