Moments of Subsets of General Equiangular Tight Frames

TL;DR

This paper proves that the moments of randomly-selected subsets of general complex ETFs converge to MANOVA moments, extending previous results for real ETFs and providing recursive formulas verified computationally.

Contribution

It extends the proof of Kesten-McKay moments to general complex ETFs and establishes recursive computation of moments matching MANOVA distributions.

Findings

Moments of subset of general ETFs converge to MANOVA moments

Recursive formulas for moments are derived and verified

Extension of previous real ETF results to complex ETFs

Abstract

This note outlines the steps for proving that the moments of a randomly-selected subset of a general ETF (complex, with aspect ratio $0<\gamma<1$) converge to the corresponding MANOVA moments. We bring here an extension for the proof of the 'Kesten-Mckay' moments (real ETF, $\gamma=1/2$) \cite{magsino2020kesten}. In particular, we establish a recursive computation of the $r$th moment, for $r = 1,2,\ldots$, and verify, using a symbolic program, that the recursion output coincides with the MANOVA moments.