Lower deviation and moderate deviation probabilities for maximum of a branching random walk

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 $M_n$ be the maximal position of individuals at the $n$-th generation. Under some mild conditions, it is known from \cite{A13} that as $n\rightarrow\infty$, $M_n-x^*n+\frac{3}{2\theta^*}\log n$ converges in law for some suitable constants $x^*$ and $\theta^*$. In this work, we investigate its moderate deviation, in other words, the convergence rates of $$\mathbb{P}\left(M_n\leq x^*n-\frac{3}{2\theta^*}\log n-\ell_n\right),$$ for any positive sequence $(\ell_n)$ such that $\ell_n=O(n)$ and $\ell_n\uparrow\infty$. As a by-product, we also obtain lower deviation of $M_n$; i.e., the convergence rate of $ \mathbb{P}(M_n\leq xn)$, for $x

Mots clés: B\"{o}ttcher case; small ball probability; Schr\"{o}der case; derivative martingale; Branching random walks; moderate deviation; lower deviation; maximal position

Domaines: Probabilités [math.PR]

Xinxin Chen, Hui He. Lower deviation and moderate deviation probabilities for maximum of a branching random walk. Annales de l'Institut Henri Poincaré, 2019, 56 (4), ⟨10.1214/20-AIHP1048⟩. ⟨hal-01874920⟩