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

By Ji Oon Lee, Kevin Schnelli. 2014

100 citations recorded by the source corpus.

We consider sample covariance matrices of the form $\mathcal{Q}=(\Sigma^{1/2}X)(\Sigma^{1/2} X)^*$, where the sample $X$ is an $M\times N$ random matrix whose entries are real independent random variables with variance $1/N$ and where $\Sigma$ is an $M\times M$ positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of $\mathcal{Q}$ when both $M$ and $N$ tend to infinity with $N/M\to d\in(0,\infty)$. For a large class of populations $\Sigma$ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of $\mathcal{Q}$ is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of $X$ are i.i.d. Gaussians or (2) that $\Sigma$ is diagonal and that the entries of $X$ have a subexponential decay.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/203464a38c0f29eec41fa575c26bb7910ee15f80

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Citations in the corpus

A Dynamical Approach to Random Matrix Theory (2017, unknown): “The edge universality was achieved in [89,94].”
A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices (2016, background): “In [33], it was proved that there exists γ0 ≡ γ0(N) depending only on πN and the aspect ratio dN such that”
Spiked separable covariance matrices and principal components (2019, background): “Gaussian X in [17, 40]; the edge universality was later proved under various moment assumptions on the entries xij [5, 14, 30, 34].”
Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, background): “See, for example, Lee and Schnelli (2014) and Knowles and Yin (2014) for some recent development.”