PreprintCC BY 4.0arXiv
Asymptotic infinitesimal freeness of covariance matrices
Abstract
We consider n× n covariance matrices M=1/nXX* where X=(xi,j) is a matrix whose entries are independent complex random variables with 𝔼(xi,j)=0 and 𝔼(|xi,j|2)=1. We derive a 1/n expansion of the mixed moments, 1/n𝔼(Tr(M(r1)⋯ M(rq))), of the form a0+a11/n+O(1/n2). This permits us to find explicit formulas for the moments and infinitesimal moments of several covariance matrices where we allow repetition. As an application of our formulas, we derive asymptotic freeness and infinitesimal freeness of independent covariance matrices under a fourth-moment condition. This generalizes previous results for the Wishart ensemble in which xi,j is complex Gaussian.
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BibTeX
@misc{george2026asymptotic,
title = {{Asymptotic infinitesimal freeness of covariance matrices}},
author = {Daniel Munoz George and Pei-Lun Tseng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18187},
eprint = {2609.18187},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18187},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Asymptotic infinitesimal freeness of covariance matrices
AU - Daniel Munoz George
AU - Pei-Lun Tseng
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18187
UR - https://arxiv.org/abs/2609.18187
AB - We consider n× n covariance matrices M=1/nXX^* where X=(x_i,j) is a matrix whose entries are independent complex random variables with 𝔼(x_i,j)=0 and 𝔼(|x_i,j|^2)=1. We derive a 1/n expansion of the mixed moments, 1/n𝔼(Tr(M^(r_1)⋯ M^(r_q))), of the form a_0+a_11/n+O(1/n^2). This permits us to find explicit formulas for the moments and infinitesimal moments of several covariance matrices where we allow repetition. As an application of our formulas, we derive asymptotic freeness and infinitesimal freeness of independent covariance matrices under a fourth-moment condition. This generalizes previous results for the Wishart ensemble in which x_i,j is complex Gaussian.
ER -