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April, 1974 Functionals of Critical Multitype Branching Processes
K. Athreya, P. Ney
Ann. Probab. 2(2): 339-343 (April, 1974). DOI: 10.1214/aop/1176996716


Let $\mathbf{Z}(t) = (\mathbf{Z}_1(t), \cdots, \mathbf{Z}_k(t)), t \geqq 0$, be a critical $k$-type, continuous time, Markov branching process. It is known that $\mathbf{Z}(t)/t$, conditioned on $\mathbf{Z}(t) \neq 0$, converges in distribution to $\mathbf{v}W$, where $\mathbf{v}$ is a vector determined by the mean matrix of the process, and $W$ is an exponentially distributed random variable. Thus if $\mathbf{\xi}$ is any fixed vector, then $(\xi \cdot \mathbf{Z}(t))/t$, conditioned on nonextinction, converges to $(\xi \cdot \mathbf{v})W$. If $\mathbf{\xi}$ is orthogonal to $\mathbf{v}$ then $t$ is not the right normalizing factor. We prove that in this case: (a) $\{(\mathbf{\xi} \cdot \mathbf{Z}(t))/(\mathbf{u} \cdot \mathbf{Z}(t))^{\frac{1}{2}} \mid \mathbf{Z}(t) \neq 0\}$ converges in distribution to a normal random variable, and (b) $\{(\mathbf{\xi} \cdot \mathbf{Z}(t))/t^{\frac{1}{2}}\mid\mathbf{Z}(t) \neq 0\}$ converges in distribution to a Laplacian random variable.


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K. Athreya. P. Ney. "Functionals of Critical Multitype Branching Processes." Ann. Probab. 2 (2) 339 - 343, April, 1974.


Published: April, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0282.60052
MathSciNet: MR356264
Digital Object Identifier: 10.1214/aop/1176996716

Primary: 60J80
Secondary: 60J85

Keywords: branching process , limit theorems

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 2 • April, 1974
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