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May, 1995 Large Deviation Rates for Branching Processes. II. The Multitype Case
K. B. Athreya, A. N. Vidyashankar
Ann. Appl. Probab. 5(2): 566-576 (May, 1995). DOI: 10.1214/aoap/1177004778


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.


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K. B. Athreya. A. N. Vidyashankar. "Large Deviation Rates for Branching Processes. II. The Multitype Case." Ann. Appl. Probab. 5 (2) 566 - 576, May, 1995.


Published: May, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0830.60082
MathSciNet: MR1336883
Digital Object Identifier: 10.1214/aoap/1177004778

Keywords: 60F , 60J , large deviations , Multitype branching processes

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.5 • No. 2 • May, 1995
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