# Harmonic Means of Wishart Random Matrices

## TL;DR

This paper analyzes the spectral properties of the harmonic mean of Wishart matrices using free probability, revealing how it compares to the arithmetic mean in operator norm for different sample sizes.

## Contribution

It introduces a free probability approach to characterize the harmonic mean of Wishart matrices and uncovers a size-dependent norm closeness phenomenon.

## Key findings

- Harmonic mean is closer to expectation than arithmetic mean for small n.
- Operator norm difference varies with the number of matrices.
- Results extend to non-identity expectation cases.

## Abstract

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.