On large deviation probabilities for empirical distribution of supercritical branching random walks with unbounded displacements

Xinxin Chen (1) , Hui He (2)

1 PSPM - Probabilités, statistique, physique mathématique
2 BNU - Beijing Normal University

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.

Mots clés: Schr\"{o}der case; large deviation; B\"{o}ttcher case; Branching random walk

Domaines: Probabilités [math.PR]

Xinxin Chen, Hui He. On large deviation probabilities for empirical distribution of supercritical branching random walks with unbounded displacements. Probability Theory and Related Fields, inPress, ⟨10.1007/s00440-018-0891-4⟩. ⟨hal-01507079v2⟩