Functionals of Critical Multitype Branching Processes

Abstract

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.