Exponential approximation and Stein's method of exchangeable pairs

TL;DR

This paper introduces a new exponential approximation result using Stein's method of exchangeable pairs, with an application to the trace of Haar-distributed unitary matrices, providing explicit error bounds.

Contribution

It develops a novel exponential approximation theorem via Stein's method of exchangeable pairs and applies it to matrix trace distributions from the Haar measure.

Findings

Derived a new exponential approximation theorem with error bounds.

Applied the theorem to the trace of Haar-distributed unitary matrices.

Provided explicit error terms for the exponential limit theorem.

Abstract

We derive a new result for exponential approximation using Stein's method of exchangeable pairs. As an application, an exponential limit theorem with error term is derived for |Tr(U)|^2, where Tr(U) denotes the trace of a matrix chosen from the Haar measure of the unitary group U(n,C).