Asymptotic properties of eigenmatrices of a large sample covariance matrix

Z. D. Bai*, H. X. Liu, Wing Keung WONG

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)

Abstract

Let Sn = 1/n XnXn where Xn = {X ij} is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let Yn(t1, t 2,σ)=√p(xn(t1) *(Sn +σI)-1xn(t2)-x n(t1)*xn(t2)m n(σ)) in which σ > 0 and mn(σ)= ∫dFyn(x)/x+σ where Fyn(x) is the Marčenko-Pastur law with parameter yn = p/n; which converges to a positive constant as n → ∞ and xn(t1) and xn(t2) are unit vectors in ℂp, having indices t 1 and t2, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence Yn(t1, t2, σ) converges weakly to a (2m + 1)-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of S n is asymptotically close to that of a Haar-distributed unitary matrix.

Original languageEnglish
Pages (from-to)1994-2015
Number of pages22
JournalAnnals of Applied Probability
Volume21
Issue number5
DOIs
Publication statusPublished - Oct 2011

Scopus Subject Areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

User-Defined Keywords

  • Central limit theorems
  • Haar distribution
  • Linear spectral statistics
  • Marčenko-Pastur law
  • Random matrix
  • Sample covariance matrix
  • Semicircular law

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