Asymptotic *-moments of some random Vandermonde matrices

TL;DR

This paper investigates the asymptotic behavior of normalized random Vandermonde matrices with uniform phase variables, revealing their convergence to a C[0,1]-valued R-diagonal element in *-distribution as matrix size grows.

Contribution

It establishes the asymptotic *-distribution of normalized Vandermonde matrices, linking their behavior to C[0,1]-valued R-diagonal elements, a novel theoretical insight.

Findings

Convergence of expected trace to a *-distribution

Identification of the limit as a C[0,1]-valued R-diagonal element

Asymptotic behavior holds as matrix size tends to infinity

Abstract

Appropriately normalized square random Vandermonde matrices based on independent random variables with uniform distribution on the unit circle are studied. It is shown that as the matrix sizes increases without bound, with respect to the expectation of the trace there is an asymptotic *-distribution, equal to that of a C[0,1]-valued R-diagonal element.