Open Access
April, 1992 On a Maximum Sequence in a Critical Multitype Branching Process
K. B. Athreya
Ann. Probab. 20(2): 746-752 (April, 1992). DOI: 10.1214/aop/1176989803


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.


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K. B. Athreya. "On a Maximum Sequence in a Critical Multitype Branching Process." Ann. Probab. 20 (2) 746 - 752, April, 1992.


Published: April, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0757.60079
MathSciNet: MR1159571
Digital Object Identifier: 10.1214/aop/1176989803

Primary: 60J80
Secondary: 60K99

Keywords: Critical branching process , Martingales , Maximum sequence

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 2 • April, 1992
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