Asymptotic local probabilities of large deviations for branching process in random environment with geometric distribution of descendants

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