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

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.