## The Annals of Probability

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

### On a Maximum Sequence in a Critical Multitype Branching Process

#### 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