The life-periods of critical branching processes with random migration

N. M. Yanev ; K. V. Mitov

Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 3, pp. 458-467

Résumé

Let $\{\xi_{ij}(t)\}_{j,t=1}^\infty$ ($i=1,2$), $\{\eta_t\}_{t=1}^\infty$ be independent integer-valued random variables with $F_i(s)=\mathbf Es^{\xi_{ij}(t)}$, $$ \mathbf P\{\eta_t=-1\}=p,\ \mathbf P\{\eta_t=0\}=q,\ \mathbf P\{\eta_t=1\}=r,\ p+q+r=1. $$ The branching process with random migration $\mu_t$ is defined by the following formula: if $\mu_t=n$ then $$ \mu_{t+1}=\sum_{j=1}^{\varphi_1(t;n)}\xi_{1j}(t+1)+\sum_{j=1}^{\varphi_2(t;n)}\xi_{2j}(t+1),\quad t=0,1,\dots, $$ where $\varphi_1(t;n)=\max\{\min\{n,n+\eta_t\},0\}$, $\varphi_2(t;n)=\max\{0,\eta_t\}$. In the critical case ($F'_1(1)=1$) we investigate the asymptotical behaviour of the probability $\mathbf P\{\tau>t\}$, $t\to\infty$, where $\tau$ is the life-period defined by the conditions $\mu_{T-1}=0$, $\mu_t>0\,(T\le t$, $\mu_{T+\tau}=0$.