Large Deviation Rates for Branching Processes. II. The Multitype Case

Abstract

Let $\{Z_n: n \geq 0\}$ be a $p$-type $(p \geq 2)$ supercritical branching process with mean matrix $M$. It is known that for any $l$ in $R^p$, $\big(\frac{l\cdot Z_n}{1\cdot Z_n} - \frac{l\cdot (Z_n M)}{1\cdot Z_n}\big) \text{and} \big(\frac{l\cdot Z_n}{1\cdot Z_n} - \frac{l\cdot v^{(1)}}{1 \cdot v^{(1)}}\big)$ converge to 0 with probability 1 on the set of nonextinction, where $v^{(1)}$ is the left eigenvector of $M$ corresponding to its maximal eigenvalue $\rho$ and 1 is the vector with all components equal to one. In this paper we study the large deviation aspects of this convergence. It is shown that the large deviation probabilities for these two sequences decay geometrically and under appropriate conditioning supergeometrically.