The Annals of Applied Probability

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

K. B. Athreya and A. N. Vidyashankar

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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


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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.