UC San Diego

We give a brief exposition of logarithmic Sobolev Inequalities (LSIs) for probability measures on R̂n, as well as some known sufficient conditions on such measures for a LSI to hold. We show that the convolution of a compactly supported probability measure on R̂n with a Gaussian measure satisfies a LSI, and look at some examples. We conclude with an application of this result by showing that the empirical law of eigenvalues of an n x n symmetric random matrix converges weakly to its mean as n [right arrow] Infinity