Concentration of a high dimensional sub-gaussian vector

Abstract

This note describes the concentration phenomenon for a high dimensional sub-gaussian vector \( X \). In the Gaussian case, for any linear operator \( Q \), it holds \( P\bigl( \| Q X \|^{2} - tr (B) > 2 \sqrt{x\, tr(B^{2})} + 2 \| B \| x \bigr) \leq e^{-x} \) and \( P\bigl( \| Q X \|^{2} - tr (B) < - 2 \sqrt{x \, tr(B^{2})} \bigr) \leq e^{-x} \) with \( B = Q \, Var(X) Q^{T} \); see \cite{laurentmassart2000}. This implies concentration of the squared norm \( \| Q X \|^{2} \) around its expectation \( E \| Q X \|^{2} = tr (B) \) provided that \( tr(B^2)/\| B \|^2 \) is sufficiently large. An extension of this result to a non-gaussian case is a nontrivial task even under sub-gaussian behavior of \( X \), especially if the entries of \( X \) cannot be assumed independent and Hanson-Wright type bounds do not apply. The results of this paper extend the Gaussian deviation bounds and support the concentration phenomenon for \( \| Q X \|^{2} \) using recent advances in Laplace approximation from \cite{SpLaplace2022} and \cite{katsevich2023tight}. The results are illustrated by the case when \( X \) is an i.i.d. sum.