## The Annals of Probability

### L1 bounds in normal approximation

Larry Goldstein

#### Abstract

The zero bias distribution W* of W, defined though the characterizing equation EWf(W)=σ2Ef'(W*) for all smooth functions f, exists for all W with mean zero and finite variance σ2. For W and W* defined on the same probability space, the L1 distance between F, the distribution function of W with EW=0 and Var(W)=1, and the cumulative standard normal Φ has the simple upper bound

F−Φ‖1≤2E|W*W|.

This inequality is used to provide explicit L1 bounds with moderate-sized constants for independent sums, projections of cone measure on the sphere S(np), simple random sampling and combinatorial central limit theorems.

#### Article information

Source
Ann. Probab., Volume 35, Number 5 (2007), 1888-1930.

Dates
First available in Project Euclid: 5 September 2007

https://projecteuclid.org/euclid.aop/1189000931

Digital Object Identifier
doi:10.1214/009117906000001123

Mathematical Reviews number (MathSciNet)
MR2349578

#### Citation

Goldstein, Larry. L 1 bounds in normal approximation. Ann. Probab. 35 (2007), no. 5, 1888--1930. doi:10.1214/009117906000001123. https://projecteuclid.org/euclid.aop/1189000931

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