The Annals of Probability

On a Maximum Sequence in a Critical Multitype Branching Process

K. B. Athreya

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Abstract

Let $\{Z_n\}$ be a $p$ type positively regular nonsingular critical branching process with mean matrix $M$. If $\nu$ is a right eigenvector of $M$ for the eigenvalue 1 and $Y_n = Z_n \cdot \nu$, and if $M_n = \max_{0\leq j\leq n}Y_j$, then it is shown that under second moments $(\log n)^{-1}E_\mathbf{i}M_n \rightarrow \mathbf{i \cdot v}$, where $E_\mathbf{i}$ denotes starting with $Z_0 = \mathbf{i}$ and $\cdot$ denotes inner product. This is an extension of the result for the single type case obtained by Athreya in 1988.

Article information

Source
Ann. Probab., Volume 20, Number 2 (1992), 746-752.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989803

Digital Object Identifier
doi:10.1214/aop/1176989803

Mathematical Reviews number (MathSciNet)
MR1159571

Zentralblatt MATH identifier
0757.60079

JSTOR
links.jstor.org

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K99: None of the above, but in this section

Keywords
Maximum sequence critical branching process martingales

Citation

Athreya, K. B. On a Maximum Sequence in a Critical Multitype Branching Process. Ann. Probab. 20 (1992), no. 2, 746--752. doi:10.1214/aop/1176989803. https://projecteuclid.org/euclid.aop/1176989803


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