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Teoriya Veroyatnostei i ee Primeneniya, 1982, Volume 27, Issue 4, Pages 667–683
(Mi tvp2404)
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This article is cited in 2 scientific papers (total in 2 papers)
Asymptotics of the extinction probability for critical general branching processes
V. A. Topčiĭ Omsk
Abstract:
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
$$
B\alpha^2-2c_i-2a\alpha=0.
$$
Received: 10.09.1980
Citation:
V. A. Topčiǐ, “Asymptotics of the extinction probability for critical general branching processes”, Teor. Veroyatnost. i Primenen., 27:4 (1982), 667–683; Theory Probab. Appl., 27:4 (1983), 715–733
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Abstract page: | 139 | Full-text PDF : | 64 | First page: | 1 |
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