For Gaussian i.i.d. data, the singular-value spectrum of the empirical cross-covariance is governed by a cubic Stieltjes equation, with simplified edge formulas in several asymptotic regimes.
(A13) reduces to: αh3z2p2 X + h2zpX (α(1 − pX ) + (1− αpX )) + h (1 − pX )(1 − αpX ) − zαp2 X + αp2 X = 0
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Distribution of singular values in large sample cross-covariance matrices
For Gaussian i.i.d. data, the singular-value spectrum of the empirical cross-covariance is governed by a cubic Stieltjes equation, with simplified edge formulas in several asymptotic regimes.