The Annals of Applied Probability

The Growth and Spread of the General Branching Random Walk

J. D. Biggins

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Abstract

A general (Crump-Mode-Jagers) spatial branching process is considered. The asymptotic behavior of the numbers present at time $t$ in sets of the form $\lbrack ta, \infty)$ is obtained. As a consequence it is shown that if $B_t$ is the position of the rightmost person at time $t, B_t/t$ converges to a constant, which can be obtained from the individual reproduction law, almost surely on the survival set of the process. This generalizes the known discrete-time results.

Article information

Source
Ann. Appl. Probab., Volume 5, Number 4 (1995), 1008-1024.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177004604

Digital Object Identifier
doi:10.1214/aoap/1177004604

Mathematical Reviews number (MathSciNet)
MR1384364

Zentralblatt MATH identifier
0859.60075

JSTOR
links.jstor.org

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Spatial spread asymptotic speed propagation rate CMJ process

Citation

Biggins, J. D. The Growth and Spread of the General Branching Random Walk. Ann. Appl. Probab. 5 (1995), no. 4, 1008--1024. doi:10.1214/aoap/1177004604. https://projecteuclid.org/euclid.aoap/1177004604


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