Affiliation:
1. Steklov Mathematical Institute of Russian Academy of Sciences , petersburg , Russia
Abstract
Abstract
We consider the branching process
Z
n
=
X
n
,
1
+
⋯
+
X
n
Z
n
−
1
$ Z_{n} =X_{n, 1} + \dotsb +X_{nZ_{n-1}} $
, in random environments
η
, where
η
is a sequence of independent identically distributedvariables, for fixed
η
the random variables X
i, j
areindependent, have the geometric distribution. We suppose that the associated random walk
S
n
=
ξ
1
+
⋯
+
ξ
n
$ S_n = \xi_1 + \dotsb + \xi_n $
has positive meanμ,0 < h<h
+satisfies the right-hand Cramer’s condition
E
exp(h
ξ
i
) < ∞ for, some h
+. Under theseassumptions, we find the asymptotic representation for local probabilities
P
(Z
n
=⌊exp(θ n)⌋) for θ ∈ [θ
1, θ
2]⊂</given−names><x> </x><surname>(μ;μ
+) and someμ
+.
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics
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