Limit theorems for a critical Galton–Watson process with migration

S. V. Nagaev   ; L. V. Han

Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 3, pp. 523-534

Résumé

The critical Galton–Watson process with immigration and emigration is investigated. We consider the population of particles which develop according to the critical Galton–Watson process with the offspring generating function $f(s)$, and at each moment $n=0,1,\dots$ either $k$ ($k=0,1,\dots$) particles immigrate in the population with the probability $p_k$ or $j$ ($j=1,\dots,m$) particles of those present at time $n$ emigrate from the population with probability $q_j$, where $m$ is a fixed natural number, $$ \sum_{k=0}^\infty p_k+\sum_{k=1}^m q_k=1,\qquad q_m>0. $$ Let $Z_n$ ($n=0,1,\dots$) be the number of particles at time $n$. We suppose that $$ Z_0=0,\qquad f'(1-)=1,\qquad\sum_{k=1}^\infty kp_k-\sum_{k=1}^m kq_k=0. $$ The following results are obtained. If $$ f(0)>0,\qquad B=1/2f''(1-)<\infty,\qquad\sum_{k=1}^\infty k^2p_k<\infty, $$ then for some $A_0\in(0,\infty)$ \begin{gather*} \mathbf P\{Z_n=0\}\sim\frac{A_0}{\log n},\quad\mathbf MZ_n\sim\frac{B_n}{\log n},\quad\mathbf DZ_n\sim\frac{2B^2n^2}{\log n}\quad(n\to\infty), \\ \lim_{n\to\infty}\mathbf P\left\{\frac{\log Z_n}{\log n}<x\right\}=x,\qquad x\in[0,1]. \end{gather*}