One concern with applying this approach to genetic data is that the entries in the matrix M do not have the Gaussian distributions expected for a Wishart matrix; instead, they correspond to the three possible genotypes at each SNP. However it is not critical that the entries in the m × n matrix M be Gaussian. Soshnikov [25] showed that the same TW limit arose if the cell entries were any distribution with high-order moments no greater than the Gaussian. The matrix X is a sum of n rank 1 matrices, and Soshnikov's result suggests that the same limit would be obtained from any probability distribution in which the columns of M are independent, isotropic (all directions are equiprobable), and such that the column norms have moments no larger than those for a column of independent Gaussian entries. In all our genetic applications, the column norms are in fact bounded, so we can expect the sample covariance matrices to behave well.
