Some Theorems for Branching Processes with Several Types of Particles

Abstract

Let us consider a branching process with a continuous time parameter. Suppose that there are n types of particles. Let $\mu _{k1} (t), \cdots \mu _{kn} (t)$ be the numbers of particles of types $T_1 , \cdots ,T_n $, respectively, generated by a unique particle of type $T_k $ in the time interval $[0,t]$. Let ${\bf a} = \| {a_{ij} } \|$ be the matrix of the first differential moments and $\lambda = \max [\operatorname{Re} \lambda _1 , \cdots ,\operatorname{Re} \lambda _n ]$, where $| {{\bf a} - \lambda _i {\bf E}} | = 0$ (${\bf E}$ is a unit matrix). Theorem 1 gives an asymptotical formula for $Q_k (t) = P\{ \sum\nolimits_{j = 1}^n {\mu_{kj} } (t) > 0\} $, when $t \to \infty $ and ${\bf a}$ is an arbitrary matrix. Theorem 2 gives the limit distribution for \[ {\bf P}\left\{ {\frac{{\mu_{k1} \left( t \right)}}{t} 0$ being a certain constant) when $t \to \infty $ and $a_{11} < 0$, \[a_{22...