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Abstract: The zero bias distribution $W^*$ of $W$, defined though the characterizingequation $\mathit{EW}fW=\sigma^2Ef-W^*$ 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 $VarW=1$, and thecumulative standard normal $\Phi$ has the simple upper bound \\VertF-\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 conemeasure on the sphere $S\ell n^p$, simple random sampling and combinatorialcentral limit theorems.

Author: Larry Goldstein

Source: https://arxiv.org/

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