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(CLTs)在概率论与统计学中有着悠久的历史。它们在构建有效的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，可在对随机变量施加不同假设的条件下证明中心极限定理。与此同时，中心极限定理的量化版本（如Berry-Esseen界）也得到并行发展。本文提出利用泛函分析中的逼近理论，导出函数期望之差的显式界。