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.