Geodesic
Limit theorems for an intermediately subcritical and a strongly subcritical branching process in a random environment
Diskretnaya Matematika, Tome 13 (2001) no. 1, pp. 132-157.

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Let $\{\xi_n\}$ be an intermediately subcritical branching process in a random environment with linear-fractional generating functions, and let $m_n^+$ be the conditional mathematical expectation of $\xi_n$ under the condition that the random environment is fixed and $\xi_n>0$. We establish the convergence of the sequence of processes $\{\xi_{[nt]}/m^+_{[nt]},\ t\in(0,1)\mid \xi_n>\nobreak0\}$ as $n\to\infty$ in the sense of finite-dimensional distributions. As a corollary, we establish the convergence of the sequence of processes $\{\ln\xi_{[nt]}/\ \sqrt n,\ t\in[0,1]\mid \xi_n>0\}$ in the sense of finite-dimensional distributions to a process expressed in terms of the Brownian meander. For a strongly subcritical branching process in a random environment $\{\xi_n\}$ with linear-fractional generating functions, we establish the convergence of the sequence $\{\xi_{[nt]},\ t\in(0,1)\mid \xi_n>0\}$ in the sense of finite-dimensional distributions to a process whose all cross-sections are independent and identically distributed. This research was supported by the Russian Foundation for Basic Research, grant 98–01–00524, and INTAS, grant 99–01317. 
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V. I. Afanasyev. Limit theorems for an intermediately subcritical and a strongly subcritical branching process in a random environment. Diskretnaya Matematika, Tome 13 (2001) no. 1, pp. 132-157. http://geodesic.mathdoc.fr/item/DM_2001_13_1_a8/

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