  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.
