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

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 $\log E\exp ⟨V^{-1}\xi ,u⟩\le |u{|}^2/2$, where $V^2\ge \mathrm{Var}(\xi )$ and $|u|\le g$ for some fixed $g$. For a linear mapping $Q$, we present an upper quantile function $z_c(B,x)$ ensuring $P(|Q\xi |>z_c(B,x))\le 3e^{-x}$ 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 $x\le x_c$, the function $z_c(B,x)$ replicates the case of a Gaussian vector $\xi$, that is, $z_c^2(B,x)=\mathrm{tr}(B)+2\sqrt{x\mathrm{tr}(B^2)}+2x|B|$. 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.