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 $\mathbb{E}[\mathbf{W}_i] = \mathbf{I}$. We demonstrate an interesting phenomenon where the harmonic mean $\mathbf{H}$ of the $n$ Wishart matrices is closer in operator norm to $\mathbb{E}[\mathbf{W}_i]$ than the arithmetic mean $\mathbf{A}$ 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.