Quadratic form of heavy-tailed self-normalized random vector with applications in $\hat{I}\pm$-heavy Mar\v cenko--Pastur law

Abstract

Let $\mathbf{x}$ be a random vector with $n$ i.i.d.\ real-valued components in the domain attraction of an $\hat{I}\pm$-stable law with $\hat{I}\pm \in (0,2)$, and let $\mathbf{y}=\mathbf{x}/\Vert \mathbf{x}{\Vert }_2$ be the associated self-normalized vector on the unit sphere. For a (possibly random) Hermitian matrix ${\mathbf{A}}_n=\left(a_{ij}^{(n)}\right)$ independent of $\mathbf{y}$, we study the asymptotic law of the quadratic form ${\mathbf{y}}^{\top }{\mathbf{A}}_n\mathbf{y}$. Building on the sharp separation between diagonal and off-diagonal contributions in this heavy-tailed setting, we show that under a mild assumption on the Frobenius norm of the off-diagonal part of ${\mathbf{A}}_n$ the limiting law is solely governed by the empirical distribution of the diagonal entries and the index $\hat{I}\pm$. More precisely, if $n^{-1}{\sum }_{i=1}^n\hat{I}{´}_{a_{ii}^{(n)}}$ converges weakly almost surely to a deterministic $\hat{I}½$, then $Q_n$ converges in distribution to a non-degenerate law $\hat{I}{¼}_{\hat{I}½,\hat{I}\pm }$ characterized through its Stieltjes transform. The law $\hat{I}{¼}_{\hat{I}½,\hat{I}\pm }$ is shown to be atom-free (provided that $\hat{I}½$ is non-degenerate) with an explicit density and tractable tail behavior. As an application in random matrix theory, we derive an implicit resolvent-based representation of the $\hat{I}\pm$-heavy Marčenko--Pastur law $H_{\hat{I}\pm ,{\hat{I}}^3}$ for heavy-tailed sample correlation matrices and prove that $H_{\hat{I}\pm ,{\hat{I}}^3}$ has no atoms except possibly at the origin. For comparison with the light-tailed setting, we also provide a Hanson--Wright-type concentration inequality for ${\mathbf{y}}^{\top }{\mathbf{A}}_n\mathbf{y}$ when the components of $\mathbf{x}$ are sub-Gaussian.