Geodesic
Local lower large deviations of strongly supercritical BPREG
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 1, pp. 1-16

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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 $\boldsymbol\eta$. We assume that $\boldsymbol\eta$ is a sequence of independent identically distributed variables and for fixed $\boldsymbol\eta$ the distribution of variables $X_{i,j}$ is geometric. 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 ${\mathbf E}\exp(h\xi_i) < \infty$ as $h^{-} for some $h^{-} < -1$. Under these assumptions, we find the asymptotic representation for local probabilities ${\mathbf P}\left( Z_n = \lfloor\exp\left(\theta n\right)\rfloor \right)$, where $\theta$ is near the boundary of the first and the second deviations zones.
Keywords: branching processes, random environment, random walk, Cramer's condition, large deviations, local theorems. 
K. Yu. Denisov. Local lower large deviations of strongly supercritical BPREG. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 1, pp. 1-16. http://geodesic.mathdoc.fr/item/SEMR_2024_21_1_a15/
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