Asymptotic local lower deviations of strictly supercritical branching process in a random environment with geometric distributions of descendants

Abstract We consider local probabilities of lower deviations for branching process Z n = X n ,1 + ⋯ + X n , Z n −1 in random environment η . We assume that η is a sequence of independent identically distributed variables and for fixed η the variables X i , j are independent and have geometric distributions. We suppose that steps ξ i of the associated random walk S n = ξ 1 + ⋯ + ξ n has positive mean and satisfies left-side Cramér condition: E exp( h ξ i ) < ∞ if h − < h < 0 for some h − < − 1. Under these assumptions we find the asymptotic of the local probabilities P ( Z n = ⌊exp( θn )⌋), n → ∞, for θ ∈ (max( m − , 0); m (− 1)) and for θ in a neighbourhood of m (− 1), where m − and m (− 1) are some constants.