Hanson-Wright inequality and sub-gaussian concentration

Abstract. In this expository note, we give a modern proof of Hanson-Wright inequality for quadratic forms in sub-gaussian random variables. We deduce a useful concentration inequality for sub-gaussian random vectors. Two examples are given to illustrate these results: a concentration of distances between random vectors and subspaces, and a bound on the norms of products of random and deterministic matrices. 1. Hanson-Wright inequality Hanson-Wright inequality is a general concentration result for quadratic forms in sub-gaussian random variables. A version of this theorem was first proved in [9, 19], however with one weak point mentioned in Remark 1.2. In this article we give a modern proof of Hanson-Wright inequality, which automatically fixes the original weak point. We then deduce a useful concentration inequality for sub-gaussian random vectors, and illustrate it with two applications. Our arguments use standard tools of high-dimensional probability. The reader unfamiliar with them may benefit from consulting the tutorial [18]. Still, we wil