Central limit theorems (CLTs) have a long history in probability and statistics. They play a fundamental role in constructing valid statistical inference procedures. Over the last century, various techniques have been developed in probability and statistics to prove CLTs under a variety of assumptions on random variables. Quantitative versions of CLTs (e.g., Berry--Esseen bounds) have also been parallelly developed. In this article, we propose to use approximation theory from functional analysis to derive explicit bounds on the difference between expectations of functions.

翻译：中心极限定理在概率论和统计学中有着悠久的历史，其在构建有效统计推断方法中发挥着基础性作用。过去一个世纪以来，概率论与统计学领域已发展出多种技术，可在不同随机变量假设条件下证明中心极限定理。与此同时，中心极限定理的定量版本（如贝里-埃森界）也得到同步发展。本文提出利用泛函分析中的逼近理论，推导函数期望之差的显式上界。