BERRY-ESSEEN BOUND AND PRECISE MODERATE DEVIATIONS FOR BRANCHING RANDOM WALKS WITH PRODUCTS OF RANDOM MATRICES

Thi Thuy Bui (1) , Ion Grama (1) , Quansheng Liu (1)

1 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique

We consider a branching random walk where particles give birth to children as a Galton-Watson process, which move in $ \mathbb R^d $ according to products of independent and identically distributed random matrices. We establish a Berry-Esseen bound and a Cramér type moderate deviation expansion for the counting measure which counts the number of particles in generation $n$ situated in a region, as $ n \rightarrow \infty $. In the proof, we construct a new martingale, and establish its uniform convergence as well as that of the fundamental martingale.

Domaines: Probabilités [math.PR]

Thi Thuy Bui, Ion Grama, Quansheng Liu. BERRY-ESSEEN BOUND AND PRECISE MODERATE DEVIATIONS FOR BRANCHING RANDOM WALKS WITH PRODUCTS OF RANDOM MATRICES. 2020. ⟨hal-02934083⟩