Limit theorems for a sequence of branching processes with immigration

I. S. Badalbaev ; A. M. Zubkov

Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 382-388

Résumé

We consider a family $Z^{(n)}(\,\cdot\,)$ of branching processes with immigration defined by a formula $$ Z^{(n)}(t)=\sum_{k\colon\theta_k^{(n)}\le t}\zeta_k^{(n)}(t-\theta_k^{(n)}), $$ where $\theta_k^{(n)}$ – the moment of immigration of k$^{\text{th}}$ particle and $\zeta_k^{(n)}(\,\cdot\,)$ – a branching process of its descendants. It is supposed that: $$ \text{i)}\quad \mathbf P\{0\le\theta_1^{(n)}\le\theta_2^{(n)}\le\dotsb,\ \lim_{k\to\infty}\theta_k^{(n)}\}=1 $$ and all finite-dimensional distributions of the processes $$ \tau^{(n)}(\alpha)=n^{-1}\sum_{k\colon\theta_k^{(n)}\le\alpha n}1 $$ converge to the corresponding finite-dimensional distrutions of a random process $T(\alpha)$, $\alpha\in[0,1]$ which is stochastically continuous at $\alpha=1$; $$ \text{ii)}\quad \mathbf Ms^{\xi_k^{(n)}(t)}=1-\frac{1-s}{1+(1-s)t\gamma}(1+\alpha_n(t;s)), $$ where $\gamma=\mathrm{const}$ and $\alpha_n(t;s)\to 0$, $n\to\infty$, uniformly in the set $\{\varepsilon n\le t\le n,\,|s|\le 1\}$ for every $\varepsilon>0$. Theorem 1. If the conditions i) and ii) are fulfilled, then $$ \lim_{n\to\infty}\mathbf M\exp\biggl\{-u\frac{Z^{(n)}(n)}{n\gamma}\biggr\}=\mathbf M\exp\biggl\{-\frac{u}{\gamma}\int_0^1\frac{dT(s)}{1+(1-s)u}\biggr\}. $$ Some generalizations are considered also.