Asymptotical local probabilities of lower deviations for branching process in random environment with geometric distributions of descendants

Abstract We consider local probabilities of lower deviations for branching process Zn=Xn,1+⋯+Xn,Zn−1 ${{Z}_{n}}={{X}_{n,1}}+\cdots +{{X}_{n,{{Z}_{n-1}}}}$in random environment η. We assume that η is a sequence of independent identically distributed random variables and for fixed environment η the distributions of variables Xi,j are geometric ones.We suppose that the associated random walk Sn=ξ1+⋯+ξn ${{S}_{n}}={{\xi }_{1}}+\cdots +{{\xi }_{n}}$has positive mean μ and satisfies left-hand Cramer’s condition Eexp(hξi)<∞ if h−