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On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence

hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorChafaï, Djalil
HAL ID: 11025
ORCID: 0000-0002-1446-1428		*
hal.structure.identifier
dc.contributor.authorTikhomirov, Konstantin*
dc.date.accessioned2018-01-27T16:14:57Z
dc.date.available2018-01-27T16:14:57Z
dc.date.issued2018
dc.identifier.issn0178-8051
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/17361
dc.language.isoenen
dc.subjectConvex body
dc.subjectRandom matrix
dc.subjectCovariance matrix
dc.subjectSingular value
dc.subjectOperator norm
dc.subjectSherman–Morrison formula
dc.subjectThin-shell inequality
dc.subjectLog-concave distribution
dc.subjectDependence
dc.subject.ddc519en
dc.titleOn the convergence of the extremal eigenvalues of empirical covariance matrices with dependence
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenConsider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge, when the dimension and the sample size both tend to infinity, to the left and right edges of the Marchenko--Pastur distribution. The assumptions are related to tails of norms of orthogonal projections. They cover isotropic log-concave random vectors as well as random vectors with i.i.d. coordinates with almost optimal moment conditions. The method is a refinement of the rank one update approach used by Srivastava and Vershynin to produce non-asymptotic quantitative estimates. In other words we provide a new proof of the Bai and Yin theorem using basic tools from probability theory and linear algebra, together with a new extension of this theorem to random matrices with dependent entries.
dc.relation.isversionofjnlnameProbability Theory and Related Fields
dc.relation.isversionofjnlvol170
dc.relation.isversionofjnlissue3-4
dc.relation.isversionofjnldate2018
dc.relation.isversionofjnlpages847-889
dc.relation.isversionofdoi10.1007/s00440-017-0778-9
dc.relation.isversionofjnlpublisherSpringer
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.relation.forthcomingouien
dc.relation.forthcomingprintouien
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dc.description.readershiprecherche
dc.description.audienceInternational
dc.relation.Isversionofjnlpeerreviewedoui
dc.date.updated2020-01-16T15:23:46Z
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