A Simple Proof of a Result of Kesten and Stigum on Supercritical Multitype Galton-Watson Branching Process

Abstract

Let $\{\mathbf{Z}_n:n \geqq 0\}$ be a supercritical $p$-type $(p \geqq 2)$ Galton-Watson branching process with offspring probability generating functions (pgf) $h_i(\mathbf{s}) i = 1,2,\cdots, p$. Assume (i) $m_{ij} \equiv \partial h_i/\partial s_j\mid_{s=1} < \infty$ for all $i$ and $j$ where $\mathbf{s} = (s_1, \cdots, s_p)$ and $\mathbf{1} = (1, 1, \cdots, 1)$, (ii) $\exists n_0 > 0 \ni$ if $M \equiv ((m_{ij}))$ then $M^{n0} \simeq 0$ (i.e. each element of $M^{n0}$ is $> 0$) and (iii) the largest real eigenvalue $\rho$ of $M$ is $> 1$. Let $\mathbf{u} \simeq 0$ and $\mathbf{v} \simeq 0$ be column vectors such that $M\mathbf{v} = \rho\mathbf{v}, \mathbf{u}'M = \rho\mathbf{u}', \mathbf{u} \cdot \mathbf{1} = 1, \mathbf{u} \cdot \mathbf{v} = 1$ where $\mathbf{u}'$ denotes transpose of $\mathbf{u}$ and $\cdot$ refers to inner product. Kesten and Stigum [6] showed (i) there always exists a nonnegative random variable $W$ such that $\mathbf{Z}_n\rho^{-n}$ converges almost surely (a.s.) to $\mathbf{u}W$ and (ii) $P(W = 0) < 1$ if and only if $E(Z_1^j \log Z_1^j\mid\mathbf{Z}_0 = e_i) < \infty$ for all $i$ and $j$ where $e_i = (\delta_{i1}, \delta_{i2}, \cdots, \delta_{ip}), \delta_{ij} = 1$ if $i = j$ and 0 if $i \neq j, Z_1^j$ is the $j$th coordinate of $\mathbf{Z}_1$. We give here a simple proof of a modified result which is exactly the same as above except that convergence a.s. is replaced by convergence in probability. We do this by showing that without any extra assumption other than the existence of $M$ the vector $(\mathbf{v}\cdot \mathbf{Z}_n)^{-1}Z_n$ converges in probability to $\mathbf{u}$ on the set of non-extinction