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

Abstract

Abstract We consider the branching process Zn=Xn,1+⋯+XnZn−1 $ Z_{n} =X_{n, 1} + \dotsb +X_{nZ_{n-1}} $, in random environmentsη, where η is a sequence of independent identically distributedvariables, for fixed η the random variables Xi, j areindependent, have the geometric distribution. We suppose that the associated random walk Sn=ξ1+⋯+ξn $ S_n = \xi_1 + \dotsb + \xi_n $ has positive meanμ,0 < h (μ;μ+) and someμ+.