The Annals of Applied Probability

Generalisations of the Bienayme-Galton-Watson Branching Process via its Representation as an Embedded Random Walk

M. P. Quine and W. Szczotka

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Abstract

We define a stochastic process $\mathscr{X} = \{X_n, n = 0,1,2, \ldots\}$ in terms of cumulative sums of the sequence $K_1, K_2, \ldots$ of integer-valued random variables in such a way that if the $K_i$ are independent, identically distributed and nonnegative, then $\mathscr{X}$ is a Bienayme-Galton-Watson branching process. By exploiting the fact that $\mathscr{X}$ is in a sense embedded in a random walk, we show that some standard branching process results hold in more general settings. We also prove a new type of limit result.

Article information

Source
Ann. Appl. Probab., Volume 4, Number 4 (1994), 1206-1222.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177004912

Digital Object Identifier
doi:10.1214/aoap/1177004912

Mathematical Reviews number (MathSciNet)
MR1304782

Zentralblatt MATH identifier
0820.60069

JSTOR
links.jstor.org

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J15 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J99: None of the above, but in this section

Keywords
Bienayme-Galton-Watson branching process embedded random walk martingale

Citation

Quine, M. P.; Szczotka, W. Generalisations of the Bienayme-Galton-Watson Branching Process via its Representation as an Embedded Random Walk. Ann. Appl. Probab. 4 (1994), no. 4, 1206--1222. doi:10.1214/aoap/1177004912. https://projecteuclid.org/euclid.aoap/1177004912


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