Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample\n Covariance Matrices with General Population

2014/09/17 by Ji Oon Lee, Lee, Ji Oon, Kevin Schnelli +1 · 2 citations
Mathematics · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.1409.4979

Abstract

We consider sample covariance matrices of the form\n\Q=(\Σ^1/2X)(\Σ^1/2 X)^*, where the sample X is an\nM\× N random matrix whose entries are real independent random variables\nwith variance 1/N and where \Σ is an M\× M positive-definite\ndeterministic matrix. We analyze the asymptotic fluctuations of the largest\nrescaled eigenvalue of \Q when both M and N tend to infinity\nwith N/M\→ d\∈(0,\∞). For a large class of populations \Σ in the\nsub-critical regime, we show that the distribution of the largest rescaled\neigenvalue of \Q is given by the type-1 Tracy-Widom distribution\nunder the additional assumptions that (1) either the entries of X are i.i.d.\nGaussians or (2) that \Σ is diagonal and that the entries of X have a\nsubexponential decay.\n

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