On large deviation probabilities for empirical distribution of branching random walks: Schröder case and Böttcher case
Résumé
Given a super-critical branching random walk on $\mathbb{R}$ started from the origin, let $Z_n(\cdot)$ be the counting measure which counts the number of individuals at the $n$-th generation located in a given set. Under some mild conditions, it is known in \cite{B90} that for any interval $A\subset \mathbb{R}$, $\frac{Z_n(\sqrt{n}A)}{Z_n(\mathbb{R})}$ converges a.s. to $\nu(A)$, where $\nu$ is the standard Gaussian measure.
In this work, we investigate the convergence rates of $$\mathbb{P}\left(\frac{Z_n(\sqrt{n}A)}{Z_n(\mathbb{R})}-\nu(A)>\Delta\right),$$
for $\Delta\in (0, 1-\nu(A))$, in both Schröder case and Böttcher case.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)