Concentration of Measure and Functional Inequalities

This thesis is devoted to the study of the concentration of measure phenomenon and its connections with functional inequalities. We focus on the relations between various types of inequalities and in the case of concentration estimates, we are mostly interested in discrete dependent random variables. In particular, we prove that Beckner inequalities with constants separated from zero as $p\to1^+$ are equivalent to the modified log Sobolev inequality. Further, we derive Sobolev type moment estimates which hold under these functional inequalities. We illustrate these results with applications to concentration of measure estimates for various stochastic models, including random permutations, zero-range processes, strong Rayleigh measures, exponential random graphs, and geometric functionals on the Poisson path space. Then, we answer an open problem posed by Mossel--Oleszkiewicz--Sen regarding relations between $p$-log-Sobolev inequalities for $p\in(0,1]$. We show that for any interval $I\subset (0,1]$, there exist $q,p\in I$, $q<p$ and a measure $\mu$ for which the $q$-log-Sobolev inequality holds, while the $p$-log-Sobolev inequality is violated. As a tool we develop certain necessary and sufficient conditions characterizing those inequalities in the case of birth-death processes on N. We also investigate concentration properties of functions of random vectors with values in the discrete cube, satisfying the stochastic covering property (SCP). Our result for SCP measures include subgaussian inequalities of bounded-difference type and their counterparts for matrix-valued setting. We also treat in detail the special case of independent Bernoulli random variables conditioned on their sum for which we obtain strengthened estimates, deriving in particular modified log-Sobolev inequalities, Talagrand's convex distance inequality and, as corollaries, concentration results for convex functions and polynomials, as well as improved estimates for matrix-valued functions. Finally, we prove a Bennett-type concentration bound for suprema of empirical processes based on sampling without replacement and a corresponding bound in the case of an arbitrary Hoeffding statistics.