Large deviations of branching process in a random environment. II
Diskretnaya Matematika, Tome 32 (2020) no. 1, pp. 135-156
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We consider the probabilities of large deviations for the branching process $ Z_n $ in a random environment, which is formed by independent identically distributed variables. It is assumed that the associated random walk $ S_n = \xi_1 + \ldots + \xi_n $ has a finite mean $ \mu $ and satisfies the Cramér condition $ E e^{h \xi_i} \infty $, $ 0 $. Under additional moment constraints on $ Z_1 $, the exact asymptotic of the probabilities $ {\mathbf P} (\ln Z_n \in [x, x + \Delta_n)) $ is found for the values $ x/n $ varying in the range depending on the type of process, and for all sequences $ \Delta_n $ that tend to zero sufficiently slowly as $ n \to \infty $. A similar theorem is proved for a random process in a random environment with immigration.
Keywords:
branching processes in random environment, large deviation probabilities, branching processes with immigration.
@article{DM_2020_32_1_a9,
author = {A. V. Shklyaev},
title = {Large deviations of branching process in a random environment. {II}},
journal = {Diskretnaya Matematika},
pages = {135--156},
publisher = {mathdoc},
volume = {32},
number = {1},
year = {2020},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/DM_2020_32_1_a9/}
}
A. V. Shklyaev. Large deviations of branching process in a random environment. II. Diskretnaya Matematika, Tome 32 (2020) no. 1, pp. 135-156. http://geodesic.mathdoc.fr/item/DM_2020_32_1_a9/