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We consider local probabilities of lower deviations for branching processZ  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 walkS  n   =   ξ  1   +  ⋯  +   ξ  n${{S}_{n}}={{\xi }_{1}}+\cdots +{{\xi }_{n}}$   has positive mean  μ  and satisfies left-hand Cramer’s conditionE   exp   (   h   ξ  i    )   &lt;  ∞   if    h  −   &lt;  h  &lt;  0$\mathbf{E}\exp \left( h{{\xi }_{i}} \right)&lt;\infty \text{ if }{{h}^{-}}&lt;h&lt;0$   for someh  −   &lt;  −  1.${{h}^{-}}&lt;-1.$   Under these assumptions, we find the asymptotic representation of local probabilitiesP    (    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  − .

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