Quadratic form of heavy-tailed self-normalized random vector with applications in $α$-heavy Mar\v cenko--Pastur law
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866910044974481408 |
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| author | Dong, Zhaorui Heiny, Johannes Yao, Jianfeng |
| author_facet | Dong, Zhaorui Heiny, Johannes Yao, Jianfeng |
| contents | Let $\mathbf{x}$ be a random vector with $n$ i.i.d.\ real-valued components in the domain attraction of an $α$-stable law with $α\in(0,2)$, 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 $α$. More precisely, if $n^{-1}\sum_{i=1}^n δ_{a^{(n)}_{ii}}$ converges weakly almost surely to a deterministic $ν$, then $Q_n$ converges in distribution to a non-degenerate law $μ_{ν,α}$ characterized through its Stieltjes transform. The law $μ_{ν,α}$ is shown to be atom-free (provided that $ν$ 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 $α$-heavy Marčenko--Pastur law $H_{α,γ}$ for heavy-tailed sample correlation matrices and prove that $H_{α,γ}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07132 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quadratic form of heavy-tailed self-normalized random vector with applications in $α$-heavy Mar\v cenko--Pastur law Dong, Zhaorui Heiny, Johannes Yao, Jianfeng Probability Statistics Theory Let $\mathbf{x}$ be a random vector with $n$ i.i.d.\ real-valued components in the domain attraction of an $α$-stable law with $α\in(0,2)$, 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 $α$. More precisely, if $n^{-1}\sum_{i=1}^n δ_{a^{(n)}_{ii}}$ converges weakly almost surely to a deterministic $ν$, then $Q_n$ converges in distribution to a non-degenerate law $μ_{ν,α}$ characterized through its Stieltjes transform. The law $μ_{ν,α}$ is shown to be atom-free (provided that $ν$ 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 $α$-heavy Marčenko--Pastur law $H_{α,γ}$ for heavy-tailed sample correlation matrices and prove that $H_{α,γ}$ 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. |
| title | Quadratic form of heavy-tailed self-normalized random vector with applications in $α$-heavy Mar\v cenko--Pastur law |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2603.07132 |