We prove a Berry–Esseen theorem, local central limit theorem and (local) large (global) moderate deviations principles for i.i.d. (uniformly) random non-uniformly expanding or hyperbolic maps with exponential first return times. Using existing results the problem is reduced to certain (Young) tower extensions, which main focus of this paper. On towers we will obtain our using contraction proper...

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 ...

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...

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...

We consider the random walk among random conductances on Z. We assume that the conductances are independent, identically distributed and uniformly bounded away from 0 and infinity. We obtain a quantitative version of the central limit theorem for this random walk, which takes the form of a Berry-Esseen estimate with speed t−1/10 for d 6 2, and speed t−1/5 for d > 3, up to logarithmic corrections.

We consider the problem of whether the canonical and microcanonical ensembles are locally equivalent for short-ranged quantum Hamiltonians of N spins arranged on a ddimensional lattices. For any temperature for which the system has a finite correlation length, we prove that the canonical and microcanonical state are approximately equal on regions containing up toO(N) spins. The proof rests on a...

In this paper, we establish optimal Berry–Esseen bounds for the generalized U-statistics. The proof is based on a new theorem exchangeable pair approach by Stein’s method under general linearity condition setting. As applications, an convergence rate of normal approximation subgraph counts in Erdos?–Rényi graphs and graphon-random graph obtained.