Concentration of a high dimensional sub-gaussian vector

TL;DR

This paper extends Gaussian deviation bounds to high-dimensional sub-gaussian vectors, demonstrating concentration of quadratic forms around their expectations using recent Laplace approximation techniques.

Contribution

It introduces new concentration bounds for sub-gaussian vectors that do not require independence of entries, expanding beyond classical Gaussian results.

Findings

Concentration of quadratic forms around their mean is established.

Extension of Gaussian deviation bounds to non-Gaussian sub-gaussian vectors.

Application to i.i.d. sums illustrates the theoretical results.

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.