Singular vector distribution of sample covariance matrices

dc.contributor.author

Ding, Xiucai

dc.date.accessioned

2019-11-21T19:41:53Z

dc.date.available

2019-11-21T19:41:53Z

dc.date.issued

2019-03

dc.date.updated

2019-11-21T19:41:52Z

dc.description.abstract

<jats:title>Abstract</jats:title><jats:p>We consider a class of sample covariance matrices of the form <jats:italic>Q</jats:italic> = <jats:italic>TXX</jats:italic><jats:italic>T</jats:italic>, where <jats:italic>X</jats:italic> = (<jats:italic>x</jats:italic><jats:sub><jats:italic>ij</jats:italic></jats:sub>) is an <jats:italic>M</jats:italic>×<jats:italic>N</jats:italic> rectangular matrix consisting of independent and identically distributed entries, and <jats:italic>T</jats:italic> is a deterministic matrix such that <jats:italic>T</jats:italic>*<jats:italic>T</jats:italic> is diagonal. Assuming that <jats:italic>M</jats:italic> is comparable to <jats:italic>N</jats:italic>, 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 <jats:italic>x</jats:italic><jats:sub><jats:italic>ij</jats:italic></jats:sub> 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 <jats:italic>x</jats:italic><jats:sub><jats:italic>ij</jats:italic></jats:sub> match those of the Gaussian random variables. Similar results hold for the left singular vectors if we further assume that <jats:italic>T</jats:italic> is diagonal.</jats:p>

dc.identifier.issn

0001-8678

dc.identifier.issn

1475-6064

dc.identifier.uri

https://hdl.handle.net/10161/19516

dc.language

en

dc.publisher

Cambridge University Press (CUP)

dc.relation.ispartof

Advances in Applied Probability

dc.relation.isversionof

10.1017/apr.2019.10

dc.title

Singular vector distribution of sample covariance matrices

dc.type

Journal article

pubs.begin-page

236

pubs.end-page

267

pubs.issue

01

pubs.organisational-group

Staff

pubs.organisational-group

Duke

pubs.organisational-group

Mathematics

pubs.organisational-group

Trinity College of Arts & Sciences

pubs.publication-status

Published

pubs.volume

51

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