Geodesic
Asymptotical local probabilities of lower deviations for branching process in random environment with geometric distributions of descendants
Diskretnaya Matematika, Tome 32 (2020) no. 3, pp. 24-37

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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 $h^{-}$ 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 $\mu^-$.
Keywords: branching processes, random environments, random walks, Cramer's condition, large deviations, local theorems. 
K. Yu. Denisov. Asymptotical local probabilities of lower deviations for branching process in random environment with geometric distributions of descendants. Diskretnaya Matematika, Tome 32 (2020) no. 3, pp. 24-37. http://geodesic.mathdoc.fr/item/DM_2020_32_3_a1/
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