DEVIATION INEQUALITIES FOR DISCRETE LOG-CONCAVE DISTRIBUTIONS

Abstract

In this thesis we explore log-concave distributions starting from the Brunn-Minkowski inequality. We discuss some of the nice properties this class of distributions has. We then show new results about discrete log-concave random variables. In particular, we investigate remarkable conjecture of Feige (2006) for the class of discrete log-concave probability distributions and prove a strengthened version. More specifically, we show that the conjectured bound holds when the random variables are independent discrete log-concave with arbitrary expectation. Finally, we present various extensions of log-concavity in discrete settings. We define the notion of discrete gamma-concave random variables and establish a localization theorem. Also, we propose a definition for discrete log-concavity in higher dimensions.