Generalizations

Abstract

Concentration inequalities under log-Sobolev inequalities are optimal in the sense that they provide a Gaussian tail for statistics that are expected to satisfy a central limit theorem. However, for that very same reason, laws satisfying a log-Sobolev inequality must have a sub-Gaussian tail. One way to weaken this hypothesis is to weaken both requirements and hypotheses, for instance to assume a weaker coercivity inequality such as a Poincaré inequality. In this section, we keep the notations of the previous section. Let us recall that a probability measure ? on R^M satisfies Poincaré’s inequality with coefficient m > 0 iff for any test function ${\rm f} \in C_b^2 (R^M )$

$$\mu _\phi (\Gamma _1 (f,{\rm }f)) \ge m\mu _\phi [(f - \mu (f))^2 ].$$