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

We consider local probabilities of lower deviations for branching process Zn = Xn,1 + ⋯ + Xn,Zn−1 in random environment η. We assume that η is a sequence of independent identically distributed variables and for fixed η the variables Xi,j are independent and have geometric distributions. We suppose that steps ξi of the associated random walk Sn = ξ1 + ⋯ + ξn has positive mean and satisfies left-side Cramér condition: E exp(hξi) < ∞ if h− < h < 0 for some h− < − 1. Under these assumptions we find the asymptotic of the local probabilities P(Zn = ⌊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. [ABSTRACT FROM AUTHOR]