PreprintCC BY 4.0arXiv
Limiting ESD for a weighted sample covariance matrix
Abstract
We study the empirical spectral distribution (ESD) of the random matrix MN=1/N∑i=1N f(|πYi|2) YiYi⊤, where π is a rank k orthogonal projector on Rp, Y1,…,YN are i.i.d. standard Gaussian vectors, and k→ ∞, p/N→ γ. We here assume that f(x)=g(x/k) for some function g discontinuous at x=1. We show that the asymptotic ESD is characterized by a Marchenko--Pastur equation, independently of k. We also show that the result extends to sub Gaussian random vectors Yi and to the case where the projector is replaced with a more general weight ϕ(Yi). Last, in the Gaussian case, we study the emergence of outliers due to the weighting.
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BibTeX
@misc{peche2026limiting,
title = {{Limiting ESD for a weighted sample covariance matrix}},
author = {Sandrine Péché},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40368},
eprint = {2610.40368},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40368},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Limiting ESD for a weighted sample covariance matrix
AU - Sandrine Péché
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40368
UR - https://arxiv.org/abs/2610.40368
AB - We study the empirical spectral distribution (ESD) of the random matrix M_N=1/N∑_i=1^N f(|πY_i|^2) Y_iY_i^⊤, where π is a rank k orthogonal projector on R^p, Y_1,…,Y_N are i.i.d. standard Gaussian vectors, and k→ ∞, p/N→ γ. We here assume that f(x)=g(x/k) for some function g discontinuous at x=1. We show that the asymptotic ESD is characterized by a Marchenko--Pastur equation, independently of k. We also show that the result extends to sub Gaussian random vectors Y_i and to the case where the projector is replaced with a more general weight ϕ(Y_i). Last, in the Gaussian case, we study the emergence of outliers due to the weighting.
ER -