Sharp deviation bounds and concentration phenomenon for the squared norm of a sub-gaussian vector

Vladimir Spokoiny

Published 2023-05-13Version 1

Let \( \Xv \) be a Gaussian zero mean vector with \( \Var(\Xv) = \BBH \). Then \( \| \Xv \|^{2} \) well concentrates around its expectation \( \dimA = \E \| \Xv \|^{2} = \tr \BBH \) provided that the latter is sufficiently large. Namely, \( \P\bigl( \| \Xv \|^{2} - \tr \BBH > 2 \sqrt{\xx \tr(\BBH^{2})} + 2 \| \BBH \| \xx \bigr) \leq \ex^{-\xx} \) and \( \P\bigl( \| \Xv \|^{2} - \tr \BBH < - 2 \sqrt{\xx \tr(\BBH^{2})} \bigr) \leq \ex^{-\xx} \); see \cite{laurentmassart2000}. This note provides an extension of these bounds to the case of a sub-gaussian vector \( \Xv \). The results are based on the recent advances in Laplace approximation from \cite{SpLaplace2022}.