On Finite Range Stable-Type Concentration

Abstract

We first study the deviation probability ${\bf P}\{f(X)-{\bf E}[f(X)]\ge x\}$, where f is a Lipschitz (for the Euclidean norm) function defined on ${\bf R}^d$ and X is an $\alpha$-stable random vector of index $\alpha \in (1,2)$. We show that this probability is upper bounded by either $e^{-cx^{\alpha/(\alpha-1)}}$ or $e^{-cx^\alpha}$ according to x taking small values or being in a finite range interval. We generalize these finite range concentration inequalities to ${\bf P}\{F-m(F)\ge x\}$ where F is a stochastic functional on the Poisson space equipped with a stable Levy measure of index $\alpha\in(0,2)$ and where $m(F)$ is a median of F.