Resolvent convergence for sample second-moment matrices with heterogeneous profiles under quadratic-form control

[2109.02644] Resolvent convergence for sample second-moment matrices with heterogeneous profiles under quadratic-form control Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2109.02644 (math) [Submitted on 6 Sep 2021 ( v1 ), last revised 15 Aug 2026 (this version, v6)] Title: Resolvent convergence for sample second-moment matrices with heterogeneous profiles under quadratic-form control Authors: Cosme Louart View a PDF of the paper titled Resolvent convergence for sample second-moment matrices with heterogeneous profiles under quadratic-form control, by Cosme Louart View PDF HTML (experimental) Abstract: We study the resolvent \(G^z=\left(\frac{1}{n}XX^{\top}-zI_p\right)^{-1}\), where \(z\in\mathbb{C}\) satisfies \(\Im(z)>0\) and \(X=(x_1,\ldots,x_n)\in\mathbb{R}^{p\times n}\) is a random matrix with independent, but not necessarily identically distributed, columns. The columns are real-valued and have finite second moments, but need not be centered. We identify a deterministic equivalent \(\tilde G^z\) through a finite-dimensional fixed-point system depending on the full second-moment profile \((\mathbb{E}[x_ix_i^{\top}])_{i\in[n]}\). Our quantitative, dimension-dependent bounds are expressed in terms of moments of the centered quadratic forms \(q_i(A):=x_i^{\top}Ax_i-\mathbb{E}[x_i^{\top}Ax_i]\), normalized either by the Hilbert--Schmidt norm or by the operator norm of \(A\). In particular, no independence between the entries of a given column is required. We prove quantitative comparison bounds between \(\operatorname{tr}(BG^z)\) and \(\operatorname{tr}(B\tilde G^z)\) in several regimes. We first consider heterogeneous profiles with uniformly bounded operator norm and bounded aspect ratio. Sharper, profile-adapted arguments then cover uniformly bounded Hilbert--Schmidt second moments without any aspect-ratio condition, a common profile for which additional operator-norm estimates are obtained, and profiles taking \(k\) pairwise commuting values, for which the Hilbert--Schmidt bounds incur a linear loss in \(k\). All probabilistic estimates are global for a fixed \(z\in\mathbb{H}\); the regime \(\Im(z)\downarrow0\) is not considered. The Hilbert--Schmidt estimate yields an explicit random-to-deterministic convergence rate and recovers the Marchenko--Pastur limit in the centered i.i.d. setting under the existence of a moment strictly larger than two. Comments: Main text 59p Subjects: Probability (math.PR) ; Machine Learning (stat.ML) MSC classes: 15A52, 60B12, 62J10 Cite as: arXiv:2109.02644 [math.PR] (or arXiv:2109.02644v6 [math.PR] for this version) https://doi.org/10.48550/arXiv.2109.02644 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Cosme Louart [ view email ] [v1] Mon, 6 Sep 2021 14:21:43 UTC (158 KB) [v2] Sun, 20 Nov 2022 07:25:11 UTC (114 KB) [v3] Sun, 8 Jan 2023 15:31:21 UTC (118 KB) [v4] Wed, 13 May 2026 07:04:28 UTC (31 KB) [v5] Fri, 5 Jun 2026 01:01:08 UTC (89 KB) [v6] Sat, 15 Aug 2026 04:39:10 UTC (46 KB) Full-text links: Access Paper: View a PDF of the paper titled Resolvent convergence for sample second-moment matrices with heterogeneous profiles under quadratic-form control, by Cosme Louart View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2021-09 Change to browse by: math stat stat.ML References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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