Analysis of the limiting spectral measure of large random matrices of the separable covariance type
By Romain Couillet, Walid Hachem. 2013
51 citations recorded by the source corpus.
Consider the random matrix $\Sigma = D^{1/2} X \widetilde D^{1/2}$ where D and $\widetilde D$ are deterministic Hermitian nonnegative matrices with respective dimensions N × N and n × n, and where X is a random matrix with independent and identically distributed centered elements with variance 1/n. Assume that the dimensions N and n grow to infinity at the same pace, and that the spectral measures of D and $\widetilde D$ converge as N, n → ∞ towards two probability measures. Then it is known that the spectral measure of ΣΣ* converges towards a probability measure μ characterized by its Stieltjes transform. In this paper, it is shown that μ has a density away from zero, this density is analytical wherever it is positive, and it behaves in most cases as $\sqrt{\vert x-a \vert}$ near an edge a of its support. In addition, a complete characterization of the support of μ is provided. Aside from its mathematical interest, the analysis underlying these results finds important applications in a certain class of s...
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/992dbc0c37a67d212f29acd5e08272c0732f0314
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In Mathematical Physics
In Statistics and Probability
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- Spiked separable covariance matrices and principal components (2019, background): “pak,m2cpakqq are the real solutions to the equations fpx,mq “ 0, and Bf Bm px,mq “ 0. (2.22) Finally, we have a1 “ Op1q, m1cpa1q P pp´rσb1q´1,0q and m2cpa1q P pp´σra 1q ´1,0q. Proof. See Section 3 of [10]. We shall call ak the spectral edges. In particular, we focus on the rightmost edge λr :“ a1. N”