Suppose we sample (autosomal) genotypes from these K populations. Assume there are Mi samples from population i, and set We suppose that the divergence of each population from a root population, as measured by FST or equivalently by divergence time on the diffusion timescale, is of order τ, which is small. What are the eigenvalues of the theoretical covariance C of the samples for the marker after our mean adjustment and normalization? Let M become large, while the relative abundance of the samples stays constant across populations. It can be shown (see the mathematical details, Theorem 3) that if B has full rank, then C has K − 1 large eigenvalues that tend to infinity with M, M − K eigenvalues that are 1 + O(τ) and one zero eigenvalue that is a structural zero, arising from the fact that our mean adjusted columns all have zero sum. We are interested in the case that τ ≪ 1 while M ≫ 1.
