Orlicz norms and concentration inequalities for β-heavy tailed random variables

Linda Chamakh (1, 2) , Emmanuel Gobet (1) , Wenjun Liu (3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 Global Markets Quantitative Research - BNP Paribas
3 X-DEP-MATHAPP - Département de Mathématiques Appliquées de l'École polytechnique

We establish a new concentration-of-measure inequality for the sum of independent random variables with β-heavy tail. This includes exponential of Gaussian distributions (a.k.a. log-normal distributions), or exponential of Weibull distributions, among others. These distributions have finite polynomial moments at any order but many not have finite α-exponential moments. We exhibit a new Orlicz norm adapted to this setting of β-heavy tails, we prove a new Talagrand inequality for the sum and a new maximal inequality. As consequence, a deviation probability of the sum from its mean is obtained.

Mots clés: heavy tails; deviation inequality; Orlicz norm; Talagrand inequality; maximal inequality; empirical process

Domaines: Statistiques [math.ST]; Probabilités [math.PR]

Linda Chamakh, Emmanuel Gobet, Wenjun Liu. Orlicz norms and concentration inequalities for β-heavy tailed random variables. 2021. ⟨hal-03175697v1⟩