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**A Berry-Esseen bound for functions of independent random variables.**
*(English)*
Zbl 0671.60016

Consider a mapping t: \(R^ n\to R\), n positive integer, and the random variable \(T=t(X_ 1,...,X_ n)\), where the X are independent variables and E T\(=0\). Under certain conditions, the distribution function of T, suitably normalized, converges to the standard Gaussian distribution.

The author gives a uniform estimation of the remainder term for the Gaussian distribution. Following an earlier used procedure he approximates T by the sum \({\mathcal T}=\sum_{1\leq j\leq n}T_ j\) of the conditional expectations \(T_ j=E(T| X_ j)\) where the \(T_ j\) are independent random variables. Assuming that the \(T_ j\) have absolute third moment, he can use the Berry-Esseen estimation of the remainder term for the Gaussian approximation of the distribution of this sum. The approximation of the distribution function of T by that of \({\mathcal T}\) adds terms to the final remainder term for the Gaussian approximation of the distribution function of T.

Applying Esseen’s smoothing lemma he finds that essentially it remains to estimate the difference \(E \exp (-itT)-E \exp (-it{\mathcal T}).\) This estimation is rather complicated and the final remainder term contains absolute moments of differences in which conditional expectations are involved. The order (in n) of this final remainder term can be estimated only in special cases.

The author gives a uniform estimation of the remainder term for the Gaussian distribution. Following an earlier used procedure he approximates T by the sum \({\mathcal T}=\sum_{1\leq j\leq n}T_ j\) of the conditional expectations \(T_ j=E(T| X_ j)\) where the \(T_ j\) are independent random variables. Assuming that the \(T_ j\) have absolute third moment, he can use the Berry-Esseen estimation of the remainder term for the Gaussian approximation of the distribution of this sum. The approximation of the distribution function of T by that of \({\mathcal T}\) adds terms to the final remainder term for the Gaussian approximation of the distribution function of T.

Applying Esseen’s smoothing lemma he finds that essentially it remains to estimate the difference \(E \exp (-itT)-E \exp (-it{\mathcal T}).\) This estimation is rather complicated and the final remainder term contains absolute moments of differences in which conditional expectations are involved. The order (in n) of this final remainder term can be estimated only in special cases.

Reviewer: H.Bergström

### MSC:

60F05 | Central limit and other weak theorems |