Distribution of singular values in large sample cross-covariance matrices

For two large matrices $\mathbf{X}$ and $\mathbf{Y}$ with Gaussian i.i.d.\ entries and dimensions $T\times N_X$ and $T\times N_Y$, respectively, we derive the probability distribution of the singular values of ${\mathbf{X}}^T\mathbf{Y}$ in different parameter regimes. This extends the Marchenko-Pastur result for the distribution of eigenvalues of empirical sample covariance matrices to singular values of empirical cross-covariances. Our results will help to establish statistical significance of cross-correlations in many data-science applications.