Concentration of a high dimensional sub-gaussian vector

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.