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 $Z_{n} = X_{n, 1} + \dotsb + X_{n, Z_{n-1}}$ in random environment $\boldsymbol\eta$. We assume that $\boldsymbol\eta$ is a sequence of independent identically distributed variables and for fixed $\boldsymbol\eta$ the variables $X_{i,j}$ are independent and have geometric distributions. We suppose that steps $\xi_i$ of the associated random walk $S_n = \xi_1 + \dotsb + \xi_n$ has positive mean and satisfies left-side Cramér condition: ${\mathbf E}\exp(h\xi_i) < \infty$ if $h^{-}<h<0$ for some $h^{-} < -1$. Under these assumptions we find the asymptotic of the local probabilities ${\mathbf P}\left( Z_n = \lfloor\exp\left(\theta n\right)\rfloor \right)$, $n\to\infty$, for $\theta \in (\max(m^{-},0);m(-1))$ and for $\theta$ in a neighbourhood of $m(-1)$, where $m^{-}$ and $m(-1)$ are some constants.