On the concentration of measure phenomenon for stable and related random vectors. (Q1879834)

The authors study the concentration of measure phenomenon for stable and related random vectors. The main result implies that if \(X\) is an \(\alpha\)-stable vector in \({\mathbb R}^d\) and \(f:{\mathbb R}^d \to {\mathbb R}\) is Lipschitz with respect to the Euclidean distance, then for all \(x>0\), \[ P \{ f(X)-m(f(X)) \geq x \} \leq 1 \wedge \frac{C(\alpha,d)}{x^\alpha}, \] where \(m(f(X))\) is a median of \(f(X)\) and the constant \(C(\alpha,d)\) is explicitly given.