On finite range stable type concentration

The purpose of these notes is to further complete our understanding of the stable concentration phenomenon, by obtaining the finite range behavior of $P(F-E[F]\geq x)$, with $F=f(X)$ where $f$ is a Lipschitz function and $X$ is a stable random vector or with $F$ a stochastic functional on the Poisson space equipped with a stable L\'evy measure.