Quadratic form of heavy-tailed self-normalized random vector with applications in $\alpha$-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 $\alpha$-stable law with $\alpha\in(0,2)$, and let $\mathbf{y}=\mathbf{x}/\|\mathbf{x}\|_2$ be the associated self-normalized vector on the unit sphere. For a (possibly random) Hermitian matrix $\mathbf{A}_n=\big(a_{ij}^{(n)}\big)$ 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 $\alpha$. More precisely, if $n^{-1}\sum_{i=1}^n \delta_{a^{(n)}_{ii}}$ converges weakly almost surely to a deterministic $\nu$, then $Q_n$ converges in distribution to a non-degenerate law $\mu_{\nu,\alpha}$ characterized through its Stieltjes transform. The law $\mu_{\nu,\alpha}$ is shown to be atom-free (provided that $\nu$ 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 $\alpha$-heavy Mar\v{c}enko--Pastur law $H_{\alpha,\gamma}$ for heavy-tailed sample correlation matrices and prove that $H_{\alpha,\gamma}$ 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.