Linear Statistics of Point Processes via Orthogonal Polynomials

TL;DR

This paper uses orthogonal polynomial techniques to analyze linear statistics in circular and Jacobi beta ensembles, establishing their distributions and proving a joint central limit theorem, with new results for the Jacobi case.

Contribution

It introduces a novel application of orthogonal polynomial methods to Jacobi beta ensembles, deriving distributional results and a joint CLT.

Findings

Distribution of linear statistics identified

Joint central limit theorem proved

New results for Jacobi beta ensembles

Abstract

For arbitrary $\beta > 0$, we use the orthogonal polynomials techniques developed by R. Killip and I. Nenciu to study certain linear statistics associated with the circular and Jacobi $\beta$ ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these results are new.