Non-uniform Berry-Esseen bound by unbounded exchangeable pairs approach

In this paper, a new technique is introduced to obtain non-uniform Berry-Esseen bounds for normal and nonnormal approximations by unbounded exchangeable pairs. This technique does not rely on the concentration inequalities developed by Chen and Shao [4,5] and can be applied to the quadratic forms and the general Curie-Weiss model.

A Peccati-Tudor type theorem for Rademacher chaoses

In this article, we prove that in the Rademacher setting, a random vector with chaotic components is close in distribution to a centered Gaussian vector, if both the maximal influence of the associated kernel and the fourth cumulant of each component is small. In particular, we recover the univariate case recently established in Döbler and Krokowski (2019). Our main strategy consists in a novel adaption of the exchangeable pairs couplings initiated in Nourdin and Zheng (2017), as well as its combination with estimates via chaos decomposition.

Exchangeable Pairs, Switchings, and Random Regular Graphs

We consider the distribution of cycle counts in a random regular graph, which is closely linked to the graph's spectral properties. We broaden the asymptotic regime in which the cycle counts are known to be approximately Poisson, and we give an explicit bound in total variation distance for the approximation. Using this result, we calculate limiting distributions of linear eigenvalue statistics for random regular graphs. Previous results on the distribution of cycle counts by McKay, Wormald, and Wysocka (2004) used the method of switchings, a combinatorial technique for asymptotic enumeration. Our proof uses Stein's method of exchangeable pairs and demonstrates an interesting connection between the two techniques.