Addendum to " Sum rules via large deviations "

TL;DR

This paper addresses a gap in a previous proof by establishing a general theorem that combines large deviation principles with convex and non-convex rate functions, enabling new results in spectral measure analysis.

Contribution

It introduces a novel theorem that merges LDPs with convex and non-convex rate functions, filling a gap in existing spectral measure theory.

Findings

Proves a general theorem combining different LDPs.

Applies the theorem to spectral matrix measures.

Extends LDP results to spectral measures on the unit circle.

Abstract

In these notes we fill a gap in a proof in Section 4 of Gamboa, Nagel, Rouault [Sum rules via large deviations, J. Funct. Anal. 270 (2016), 509-559]. We prove a general theorem which combines a LDP with a convex rate function and a LDP with a non-convex one. This result will be used to prove LDPs for spectral matrix measures and for spectral measures on the unit circle.