Another look at the moment method for large dimensional random matrices

Arup Bose, Arnab Sen

Research output: Contribution to journalArticlepeer-review

35 Scopus citations

Abstract

The methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices includes the well known moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample variance covariance matrix. In a recent article Bryc, Dembo and Jiang (2006) [7] establish the LSD for the random Toeplitz and Hankel matrices using the moment method. They perform the necessary counting of terms in the trace by splitting the relevant sets into equivalent classes and relating the limits of the counts to certain volume calculations. We build on their work and present a unified approach. This helps provide relatively short and easy proofs for the LSD of common matrices while at the same time providing insight into the nature of different LSD and their interrelations. By extending these methods we are also able to deal with matrices with appropriate dependent entries .

Original languageEnglish (US)
Pages (from-to)588-628
Number of pages41
JournalElectronic Journal of Probability
Volume13
DOIs
StatePublished - Jan 1 2008

Keywords

  • Almost sure convergence
  • Bounded Lipschitz metric
  • Catalan numbers
  • Circulant matrix
  • Convergence in distribution
  • Eigenvalues
  • Hankel matrix
  • Large dimensional random matrix
  • Limiting spectral distribution
  • Linear process
  • M dependent sequence
  • Marčenko-Pastur law
  • Moment method
  • Palindromic matrix
  • Random probability
  • Reverse circulant matrix
  • Sample variance covariance matrix
  • Semicircular law
  • Spectral distribution and density
  • Symmetric circulant matrix
  • Toeplitz matrix
  • Volume method
  • Wigner matrix

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