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 ${\mathbf{W}}_i$ whose limiting aspect ratio is $\gamma \in (0,1)$ when $$\mathbb{E}$[$\mathbf{W}$_i] = $\mathbf{I}$$\mathrm{.}Wedemonstrateaninterestingphenomenonwheretheharmonicmean$$\mathbf{H}$$ofthe$n$Wishartmatricesiscloserinoperatornormto$$\mathbb{E}$[$\mathbf{W}$_i]$thanthearithmeticmean$$\mathbf{A}$$forsmall$n$, 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.