|
This article is cited in 11 scientific papers (total in 11 papers)
Limit theorems for an intermediately subcritical and a strongly subcritical branching process in a random environment
V. I. Afanasyev
Abstract:
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.
Received: 20.01.2000
Citation:
V. I. Afanasyev, “Limit theorems for an intermediately subcritical and a strongly subcritical branching process in a random environment”, Diskr. Mat., 13:1 (2001), 132–157; Discrete Math. Appl., 11:2 (2001), 105–131
Linking options:
https://www.mathnet.ru/eng/dm270https://doi.org/10.4213/dm270 https://www.mathnet.ru/eng/dm/v13/i1/p132
|
|