Deviation bounds for the norm of a random vector under exponential moment conditions with applications

TL;DR

This paper extends deviation bounds for the norm of random vectors with exponential moment conditions, revealing a phase transition in tail behavior and applications to Bernoulli sums and covariance estimation.

Contribution

It introduces new bounds under exponential moment conditions, generalizing Hanson-Wright inequality and demonstrating a phase transition in tail probabilities.

Findings

Derived phase transition in tail bounds for vectors with exponential moments.

Provided explicit quantile functions for linear transformations of such vectors.

Applied results to Bernoulli sums and covariance matrix estimation.

Abstract

Hanson-Wright inequality provides a powerful tool for bounding the norm $|\xi|$ of a centered stochastic vector $\xi$ with sub-gaussian behavior. This paper extends the bounds to the case when $\xi$ only has bounded exponential moments of the form $\log E \exp \langle V^{-1} \xi,u \rangle \leq |u|^2/2$, where $V^2 \geq \mathrm{Var}(\xi)$ and $|u| \leq g$ for some fixed $g$. For a linear mapping $Q$, we present an upper quantile function $z_{c}(B,x)$ ensuring $P(| Q \xi | > z_{c}(B,x)) \leq 3 e^{-x}$ with $B = Q \, V^2 Q^{T}$. The obtained results exhibit a phase transition effect: with a value $x_{c}$ depending on $g$ and $B$, for $x \leq x_{c}$, the function $z_{c}(B,x)$ replicates the case of a Gaussian vector $\xi$, that is, $z_{c}^2 (B,x) = {\rm tr}(B) + 2 \sqrt{x {\rm tr}(B^2)} + 2 x |B|$. For $x > x_{c}$, the function $z_{c}(B,x)$ grows linearly in $x$. The results are specified to the case of Bernoulli vector sums and to covariance estimation in Frobenius norm.