Asymptotic local probabilities of large deviations for branching process in random environment with geometric distribution of descendants

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 +.