Extreme eigenvalues of sparse, heavy tailed random matrices

Antonio Auffinger*, Si Tang

*Corresponding author for this work

Research output: Contribution to journalArticle

5 Scopus citations

Abstract

We study the statistics of the largest eigenvalues of p×p sample covariance matrices Σp,n=Mp,nMp,n when the entries of the p×n matrix Mp,n are sparse and have a distribution with tail t−α, α>0. On average the number of nonzero entries of Mp,n is of order nμ+1, 0≤μ≤1. We prove that in the large n limit, the largest eigenvalues are Poissonian if α<2(1+μ−1) and converge to a constant in the case α>2(1+μ−1). We also extend the results of Benaych-Georges and Péché (2014) in the Hermitian case, removing restrictions on the number of nonzero entries of the matrix.

Original languageEnglish (US)
Pages (from-to)3310-3330
Number of pages21
JournalStochastic Processes and their Applications
Volume126
Issue number11
DOIs
StatePublished - Nov 1 2016

Keywords

  • Eigenvalue distribution
  • Heavy tail
  • Random matrices
  • Sparse

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics

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