Moderate deviations via cumulants

TL;DR

This paper establishes moderate deviation principles for a broad class of random variables using cumulant bounds, with applications to random graphs, U-statistics, and random matrix theory.

Contribution

It introduces a general method for proving moderate deviations based on cumulant bounds and applies it to diverse probabilistic models.

Findings

Moderate deviation principles are proved for dependency graphs and subgraph counts.

Results include deviations for characteristic polynomials of random matrices.

Applications extend to eigenvalue counts in various random point fields.

Abstract

The purpose of the present paper is to establish moderate deviation principles for a rather general class of random variables fulfilling certain bounds of the cumulants. We apply a celebrated lemma of the theory of large deviations probabilities due to Rudzkis, Saulis and Statulevicius. The examples of random objects we treat include dependency graphs, subgraph-counting statistics in Erd\H{o}s-R\'enyi random graphs and $U$-statistics. Moreover, we prove moderate deviation principles for certain statistics appearing in random matrix theory, namely characteristic polynomials of random unitary matrices as well as the number of particles in a growing box of random determinantal point processes like the number of eigenvalues in the GUE or the number of points in Airy, Bessel, and $\sin$ random point fields.