We obtain rates of convergence in limit theorems of partial sums Sn for certain sequences of dependent, identically distributed random variables, which arise naturally in statistical mechanics, in particular, in the context of the Curie-Weiss models. Under appropriate assumptions there exists a real number α, a positive real number μ, and a positive integer k such that (Sn − nα)/n converges wea...

Applying Stein's method, an inductive technique and size bias coupling yields a Berry–Esseen theorem for normal approximation without the usual restriction that the coupling be bounded. The theorem is applied to counting the number of vertices in the Erd˝ os–Rényi random graph of a given degree. 1. Introduction. We present a new Berry–Esseen theorem for sums Y of dependent variables by combinin...

In 2001, Chen and Shao gave the nonuniform estimation of the rate of convergence in Berry-Esseen theorem for independent random variables via Stein-Chen-Shao method. The aim of this paper is to obtain a constant in Chen-Shao theorem, where the random variables are not necessarily identically distributed and the existence of their third moments are not assumed. The bound is given in terms of tru...

This paper is a short exposition of Stein’s method normal approximation from my personal perspective. It focuses mainly on the characterization distribution and construction Stein identities. Through examples, it provides glimpses into many approaches to constructing identities diverse applications mathematical problems. also includes anecdotes historical interest, including how discovered his ...

We obtain a sharp estimate of the speed convergence in Boolean central limit theorem for measures with finite sixth moment. The main tool is quantitative version Stieltjes-Perron inversion formula.