A geometric property of the sample mean and residuals

For a sample (x1,…,xn)(x1,…,xn) from a population with finite second moment it is proved that the angle between the sample mean x̄ and the subspace generated by the residuals x1−x̄,…,xn−x̄ monotonically decreases as nn increases and thus has a limit that, in the regular case, involves the Fisher information and looks rather elegant. It has a close connection to the estimation of a location parameter. Some numerical examples illustrate the result.