Local lower large deviations of strongly supercritical BPREG

K. Yu. Denisov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 1, pp. 1-16

Résumé

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.

Détail

Keywords: branching processes, random environment, random walk, Cramer's condition, large deviations, local theorems.