Concentration Properties of Restricted Measures with Applications to Non-Lipschitz Functions

Abstract

We show that for any metric probability space $(M,d,\mu )$ with a subgaussian constant ${\sigma }^2(\mu )$ and any set $A\subset M$ we have ${\sigma }^2({\mu }_A)\le c\log \left(e/\mu (A)\right){\sigma }^2(\mu )$, where ${\mu }_A$ is a restriction of $\mu$ to the set $A$ and $c$ is a universal constant. As a consequence we deduce concentration inequalities for non-Lipschitz functions.

18 pages