Concentration of a high dimensional sub-gaussian vector (2305.07885v5)

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.