Fine asymptotics for models with Gamma type moments

TL;DR

This paper develops detailed asymptotic results for random variables with Gamma type moments, including random determinants from various matrix ensembles and geometric volumes, using mod-$$ convergence to extend classical limit theorems.

Contribution

It introduces a unified framework for fine asymptotics of Gamma moment models, covering classical and non-classical random matrix ensembles and geometric quantities.

Findings

Extended limit theorems for Gamma moment models

Berry-Esseen bounds and deviation principles established

Asymptotics for volumes of high-dimensional simplices

Abstract

The aim of this paper is to give fine asymptotics for random variables with moments of Gamma type. Among the examples we consider are random determinants of Laguerre and Jacobi beta ensembles with varying dimensions (the number of observed variables and the number of measurements vary and may be different). In addition to the Dyson threefold way of classical random matrix models (GOE, GUE, GSE), we study random determinants of random matrices of the so-called tenfold way, including the Bogoliubov-de Gennes and chiral ensembles from mesoscopic physics. We show that fixed-trace matrix ensembles can be analysed as well. Finally, we add fine asymptotics for the $p(n)$-dimensional volume of the simplex with $p(n)+1$ points in ${\Bbb R}^n$ distributed according to special distributions, which is strongly correlated to Gram matrix ensembles. We use the framework of mod-$\varphi$ convergence to obtain extended limit theorems, Berry-Esseen bounds, precise moderate deviations, large and moderate deviation principles as well as local limit theorems. The work is especially based on the recent work of Dal Borgo, Hovhannisyan and Rouault.