Modified log-Sobolev inequalities and two-level concentration
We consider a generic modified logarithmic Sobolev inequality (mLSI) of the form Ent(mu)(e(f)) <= rho/2 E-mu, e(f) Gamma(f )(2) for some difference operator Gamma, and show how it implies two-level concentration inequalities akin to the Hanson-Wright or Bernstein inequality. This can be applied to the continuous (e. g. the sphere or bounded perturbations of product measures) as well as discrete setting (the symmetric group, finite measures satisfying an approximate tensorization property, ... ). Moreover, we use modified logarithmic Sobolev inequalities on the symmetric group S-n and for slices of the hypercube to prove Talagrand's convex distance inequality, and provide concentration inequalities for locally Lipschitz functions on S-n. Some examples of known statistics are worked out, for which we obtain the correct order of fluctuations, which is consistent with central limit theorems.
18
1
855-885
855-885
Impa