LIMIT OF THE SMALLEST EIGENVALUE OF A LARGE DIMENSIONAL SAMPLE COVARIANCE MATRIX

Abstract

Advances in Statistics, pp. 108-127 (2008) No AccessLIMIT OF THE SMALLEST EIGENVALUE OF A LARGE DIMENSIONAL SAMPLE COVARIANCE MATRIXZ. D. BAI and Y. Q. YINZ. D. BAIDepartment of Statistics, Temple University, Philadelphia, Pennsylvania 19122, USA and Y. Q. YINDepartment of Mathematics, University of Massachusetts, Lowell, Lowell, Massachusetts 01854, USAhttps://doi.org/10.1142/9789812793096_0012Cited by:5 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: In this paper, the authors show that the smallest (if p ≤ n) or the (p - n + 1)-th smallest (if p > n) eigenvalue of a sample covariance matrix of the form (1/n)XX' tends almost surely to the limit as n → ∞ and p/n → y ∈ (0, ∞), where X is a p × n matrix with iid entries with mean zero, variance 1 and fourth moment finite. Also, as a by-product, it is shown that the almost sure limit of the largest eigenvalue is , a known result obtained by Yin, Bai and Krishnaiah. The present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices. Keywords: Random matrixsample covariance matrixsmallest eigenvalue of a random matrixspectral radiusAMSC: Primary 60F15, secondary 62F99 FiguresReferencesRelatedDetailsCited By 5On Estimating Rank-One Spiked Tensors in the Presence of Heavy Tailed ErrorsArnab Auddy and Ming Yuan1 Dec 2022 | IEEE Transactions on Information Theory, Vol. 68, No. 12Second Order Central Moment Estimation of Single and Multiple Scattering Intensities in Full-field Reflective Tissue Imaging under Coherent IlluminationPeng Miao and Cheng Wang25 Apr 2022Algorithmic Analysis and Statistical Estimation of SLOPE via Approximate Message PassingZhiqi Bu, Jason M. Klusowski, Cynthia Rush and Weijie J. Su1 Jan 2021 | IEEE Transactions on Information Theory, Vol. 67, No. 1Space-Time Adaptive Detection at Low Sample SupportBenjamin D. Robinson, Robert Malinas and Alfred O. Hero1 Jan 2021 | IEEE Transactions on Signal Processing, Vol. 69Detecting changes in the second moment structure of high-dimensional sensor-type data in a K -sample settingNils Mause and Ansgar Steland6 January 2021 | Sequential Analysis, Vol. 39, No. 3 Advances in StatisticsMetrics History KeywordsRandom matrixsample covariance matrixsmallest eigenvalue of a random matrixspectral radiusPDF download

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