Central Limit Theorems and Approximation Theory

Music City Center 

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. We provide bounds on the difference between functions of random variables using level sets of functions. Using classical uniform and non-uniform Berry–Esseen bounds for univariate random variables. The resulting bounds can be applied to single-layer neural networks and functions on [-1,1]^d with finite weighted norm integrable Fourier transform. These functions belong to the functions in Barron space. Unlike the classical bounds that depend on the oscillation function of f, our bounds do not have an explicit dimension dependence.

multidimensional

central limit theorem

Berry-Esseen bound

dependence on dimension

dependence on function 

Section on Statistics and Data Science Education