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

Abstract

Abstract We consider local probabilities of lower deviations for branching process Z n = X n , 1 + ⋯ + X n , Z n − 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 X i,j are geometric ones.We suppose that the associated random walk S n = ξ 1 + ⋯ + ξ n ${{S}_{n}}={{\xi }_{1}}+\cdots +{{\xi }_{n}}$ has positive mean μ and satisfies left-hand Cramer’s condition E exp ( h ξ i ) < ∞  if  h − < h < 0 $\mathbf{E}\exp \left( h{{\xi }_{i}} \right)<\infty \text{ if }{{h}^{-}}<h<0$ for some h − < − 1. ${{h}^{-}}<-1.$ Under these assumptions, we find the asymptotic representation of local probabilities P ( Z n = ⌊ exp ( θ n ) ⌋ )  for  θ ∈ [ θ 1 , θ 2 ] ⊂ ( μ − ; μ ) $\mathbf{P}\left( {{Z}_{n}}=\left\lfloor \exp (\theta n) \right\rfloor \right)\text{ for }\theta \in \left[ {{\theta }_{1}},{{\theta }_{2}} \right]\subset \left( {{\mu }^{-}};\mu \right)$ for some non-negative μ −.