Asymptotics of the extinction probability for critical general branching processes

V. A. Topčiǐ

Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 4, pp. 667-683

Résumé

Let the Cramp–Mode–Jagers process $\xi(n)$ is defined by the random variable $\eta$ (life–length) and by the point process $N(t)$ (reproduction of individual). It is proved that if $\mathbf MN(\infty)=1$, $0<\mathbf MN(\infty)(N(\infty)-1)=B<\infty$, $$ 0<\int_0^\infty td\,\mathbf MN(t)=a<\infty,\qquad c_1=\varliminf_{t\to\infty}t^2\mathbf P\{\eta>t\},\qquad c_2=\varlimsup_{t\to\infty}t^2\mathbf P\{\eta>t\}, $$ then $$ \alpha_1\le\varliminf_{t\to\infty}t\mathbf P\{\xi(t)>0\}\le \varlimsup_{t\to\infty}t\mathbf P\{\xi(t)>0\}\le\alpha_2 $$ where $\alpha_i$ are the solutions of equations $$ B\alpha^2-2c_i-2a\alpha=0. $$