Variations and extensions of the Gaussian concentration inequality

Daniel J. Fresen

Published 2018-12-28Version 1

We use and modify the Gaussian concentration inequality to prove a variety of concentration inequalities for a wide class of functions and measures on $\mathbb{R}^{n}$, typically involving independence, various types of decay (including exponential, Weibull $0<q<1$, and polynomial) and the distribution of the gradient. We typically achieve sub-Gaussian bounds that match the variance up to a parameter independent of $n$. These methods also yield significantly shorter and simpler proofs (sometimes with improvements) of known results, such as the distribution of linear combinations of Weibull variables, Gaussian concentration of the $\ell_{p}^{n}$ norm, and the distribution of the $\ell_{p}^{n}$ norm on the $\ell_{q}^{n}$ ball. We prove sub-Gaussian bounds for linear combinations of random variables with finite $r$-moments ($r>2$) that go deeper into the tails than does the weighted Berry-Esseen inequality. We use these methods to study random sections of convex bodies, proving a variation of Milman's general Dvoretzky theorem for non-Gaussian random matricies with i.i.d. entries.