On Finite Range Stable-Type Concentration

We first study the deviation probability $P\{f(X)-E[f(X)]\ge x\}$, where $f$ is a Lipschitz (for the Euclidean norm) function defined on $R^d$ 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 $P\{F-m(F)\ge x\}$ where $F$ is a stochastic functional on the Poisson space equipped with a stable Lйvy measure of index $\alpha\in(0,2)$ and where $m(F)$ is a median of $F$.