## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 5, Number 2 (1995), 566-576.

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

K. B. Athreya and A. N. Vidyashankar

#### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 5, Number 2 (1995), 566-576.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177004778

**Digital Object Identifier**

doi:10.1214/aoap/1177004778

**Mathematical Reviews number (MathSciNet)**

MR1336883

**Zentralblatt MATH identifier**

0830.60082

**JSTOR**

links.jstor.org

**Keywords**

60J 60F Large deviations multitype branching processes

#### Citation

Athreya, K. B.; Vidyashankar, A. N. Large Deviation Rates for Branching Processes. II. The Multitype Case. Ann. Appl. Probab. 5 (1995), no. 2, 566--576. doi:10.1214/aoap/1177004778. https://projecteuclid.org/euclid.aoap/1177004778

#### See also

- Part I: K. B. Athreya. Large Deviation Rates for Branching Processes--I. Single Type Case. Ann. Appl. Probab., Volume 4, Number 3 (1994), 779--790.Project Euclid: euclid.aoap/1177004971