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Über eine globale Fehlerabschätzung im zentralen Grenzwertsatz. (On a global error estimate in the central limit theorem). (German) Zbl 0606.60031

Let \((X_ k)_{k=1,2,...,n}\) be a sequence of independent random variables with mean \(EX_ k=0\) and \(0<B^ 2_ n=\sum^{n}_{k=1}EX^ 2_ k<\infty\). Write \(F_ n(x)=P(X_ 1+X_ 2+...+X_ n<x)\) and \(\Phi (x)=(2\pi)^{-1/2}\int^{x}_{- \infty}e^{-u^ 2/2}du\). Then: \[ \delta_{np}=(\int^{\infty}_{- \infty}| F_ n(xB_ n)-\Phi (x)|^ pdx)^{1/p}\leq L(K/L)^{1/p}\sum^{n}_{k=1}E \min (1,| X_ k| /B_ n)X^ 2_ k/B^ 2_ n,\quad 1\leq p\leq \infty, \] where \(L<3.51\) and \(K<33.88\) (It is possible to show \(K<33.40)\). This error estimate is a sharpening of a result given by R. V. Erickson [Ann. Probab. 1, 497-503 (1973; Zbl 0292.60040)].

MSC:

60F05 Central limit and other weak theorems
60E10 Characteristic functions; other transforms
60G50 Sums of independent random variables; random walks

Citations:

Zbl 0292.60040
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