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Short Communications
Limit theorems for a sequence of branching processes with immigration
I. S. Badalbaeva,
A. M. Zubkovb a Taškent
b Moscow
Abstract:
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.
Received: 27.04.1982