A Remark on Sudakov Minoration for Chaos

Abstract

Consider a set A of symmetric n×n matrices a = (a i,j)i,j≤n. Consider an independent sequence \((g_i)_i \leq n\) of standard normal random variables, and let \(M=E\sup_{a\in A} \left |\sum_{i,j \leq n} a_{i,j}g_ig_j \right |\). Denote by N 2(A,β) the smallest number of balls of radius β for the l2 norm of \({{\mathbb{R}}^{{{{n}^{2}}}}}\) needed to cover A. We show that if the operator norm of each element of A is ≤ α, then α log N 2 (A, K√Mα) ≤ KM, where K is a universal constant. We show that this inequality provides a link between two of this author’s previous results on the topic.