Critical branching processes evolving in a unfavorable random environment

Abstract Let Z n, n = 0, 1, 2, … be a critical branching process in a random environment, and S n, n = 0, 1, 2, … be its associated random walk. It is known that if the increments of this random walk belong (without centering) to the domain of attraction of a stable law, then there exists a regularly varying at infinity sequence a 1, a 2, … such that conditional distributions P (S n a n ≤ x | Z n > 0), x ∈ (− ∞, + ∞), array P (S₍a₍ x| Z₍ 0), x (-, +), array converge weakly to the distribution of strictly positive proper random variable. In this paper we add to this result the description of the asymptotic behavior of the probability P (Z n > 0, S n ≤ φ (n) ), array P (Z₍ 0, S₍ (n) ), array where φ (n) → ∞ for n → ∞ in such a way that φ (n) = o (a n).