$L^1$ bounds in normal approximation

TL;DR

This paper establishes a simple bound on the $L^1$ distance between a standardized random variable's distribution and the normal distribution using the zero bias transformation, with applications to sums, projections, sampling, and combinatorial CLTs.

Contribution

It introduces an explicit $L^1$ bound in normal approximation based on the zero bias distribution, applicable to various probabilistic models.

Findings

Provides a simple upper bound for $L^1$ distance in normal approximation.

Applies the bound to sums, projections, sampling, and combinatorial CLTs.

Offers moderate-sized constants for practical bounds.

Abstract

The zero bias distribution $W^*$ of $W$, defined though the characterizing equation $\mathit{EW}f(W)=\sigma^2Ef'(W^*)$ for all smooth functions $f$, exists for all $W$ with mean zero and finite variance $\sigma^2$. For $W$ and $W^*$ defined on the same probability space, the $L^1$ distance between $F$, the distribution function of $W$ with $\mathit{EW}=0$ and $Var(W)=1$, and the cumulative standard normal $\Phi$ has the simple upper bound \[\Vert F-\Phi\Vert_1\le2E|W^*-W|.\] This inequality is used to provide explicit $L^1$ bounds with moderate-sized constants for independent sums, projections of cone measure on the sphere $S(\ell_n^p)$, simple random sampling and combinatorial central limit theorems.