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

We consider local probabilities of lower deviations for branching process $Z_{n} = X_{n, 1} + \dotsb + X_{n, Z_{n-1}}$ in random environment $\eta$. We assume that $\eta$ is a sequence of independent identically distributed random variables and for fixed environment $\boldsymbol\eta$ the distributions of variables $X_{i,j}$ are geometric ones. We suppose that the associated random walk $S_n = \xi_1 + \dotsb + \xi_n$ has positive mean $\mu$ and satisfies left-hand Cramer's condition ${\mathbf E}\exp(h\xi_i) < \infty$ if $h^{-}<h<0$ for some $h^{-} < -1$. Under these assumptions, we find the asymptotic representation of local probabilities ${\mathbf P}\left( Z_n = \lfloor\exp\left(\theta n\right)\rfloor \right)$ for $\theta \in [\theta_1,\theta_2] \subset (\mu^-;\mu)$ for some non-negative $\mu^-$.