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 \| ξ \| of a centered stochastic vector ξ with independent entries and sub-gaussian behavior. This paper extends the bounds to the case when ξ only has bounded exponential moments of the form log E exp V⁻¹ ξ,u ≤ \| u \|²/2, where V² ≥ Var(ξ) and \| u \| ≤ g for some fixed g. For a linear mapping Q, we present an upper quantile function zc(B,x) ensuring P(\| Q ξ \| > zc(B,x)) ≤ 3 e⁻ˣ with B = Q \, V² QT. The obtained results exhibit a phase transition effect: with a value xc depending on g and B, for x ≤ xc, the function zc(B,x) replicates the case of a Gaussian vector ξ, that is, zc²(B,x) = tr(B) + 2 √{x {tr}(B²)} + 2 x \| B \| $. For $ x > xc $, the function $ zc(B,x) $ grows linearly in $ x $. The results are specified to the case of Bernoulli vector sums and to covariance estimation in Frobenius norm.