Some Theorems for Branching Processes with Several Types of Particles

A. A. Savin ; V. P. Chistyakov

Teoriâ veroâtnostej i ee primeneniâ, Tome 7 (1962) no. 1, pp. 95-104

Résumé

Let us consider a branching process with a continuous time parameter. Suppose that there are $n$ types of particles. Let $\mu_{k1}(t),\dots,\mu_{kn}(t)$ be the numbers of particles of types $T_1,\dots,T_n$, respectively, generated by a unique particle of type $T_k$ in the time interval $[0,t]$. Let ${\mathbf a}=\|{a_{ij}}\|$ be the matrix of the first differential moments and $\lambda=\max[\operatorname{Re}\lambda_1,\dots,\operatorname{Re}\lambda_n]$, where $|{{\mathbf a}-\lambda _i{\mathbf E}}|=0$ (${\mathbf E}$ 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 ${\mathbf a}$ is an arbitrary matrix. Theorem 2 gives the limit distribution for $${\mathbf P}\left\{{\frac{{\mu_{k1}\left(t\right)}}{t},\frac{{\mu_{k2}\left(t\right)}}{t},\frac{{B\mu_{k3}\left(t\right)}}{t}}\right\}$$ ($\beta>0$ being a certain constant) when $t\to\infty $ and $a_{11}0$, $$a_{22}\leq0,a_{33}=0,a_{12}>0,a_{13}\geq0,a_{23}>0,a_{ij}=0,i>j.$$