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Singular vector distribution of sample covariance matrices

Part of: Equilibrium statistical mechanics Special matrices

Published online by Cambridge University Press:  22 July 2019

Xiucai Ding
Xiucai Ding*
Affiliation:
University of Toronto
*
*Postal address: Department of Statistical Sciences, University of Toronto, Sidney Smith Hall, 100 St. George Street, Toronto, ON M5S 3G3, Canada.
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Abstract

We consider a class of sample covariance matrices of the form Q = TXX*T*, where X = (xij) is an M×N rectangular matrix consisting of independent and identically distributed entries, and T is a deterministic matrix such that T*T is diagonal. Assuming that M is comparable to N, we prove that the distribution of the components of the right singular vectors close to the edge singular values agrees with that of Gaussian ensembles provided the first two moments of xij coincide with the Gaussian random variables. For the right singular vectors associated with the bulk singular values, the same conclusion holds if the first four moments of xij match those of the Gaussian random variables. Similar results hold for the left singular vectors if we further assume that T is diagonal.

Keywords

Random matrix theorysingular vector distributiondeformed Marcenko–Pastur law

MSC classification

Primary: 15B52: Random matrices
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 1 , March 2019 , pp. 236 - 267
Copyright
© Applied Probability Trust 2019 

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Footnotes

The supplementary material for this article can be found at http://doi.org/10.1017/apr.2019.10.

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