  In this note, we claim that diagonal scaling of a sample covariance matrix is
asymptotically inconsistent if the ratio of the dimension to the sample size
converges to a positive constant, where population is assumed to be Gaussian
with a spike covariance model. Our non-rigorous proof relies on the replica
method developed in statistical physics. In contrast to similar results known
in literature on principal component analysis, the strong inconsistency is not
observed. Numerical experiments support the derived formulas.
