Harmonic means of Wishart random matrices

We use free probability to compute the limiting spectral properties of the harmonic mean of n i.i.d. Wishart random matrices W-i whose limiting aspect ratio is gamma is an element of (0, 1) when E[W-i] = I. We demonstrate an interesting phenomenon where the harmonic mean H of the n Wishart matrices is closer in operator norm to E[W-i] than the arithmetic mean A for small n, after which the arithmetic mean is closer. We also prove some results for the general case where the expectation of the Wishart matrices are not the identity matrix.