independent identically distributed random variables
Some nonuniform estimate in the central limit theorem
Let \(\xi_1,\ldots,\xi_{n},\ldots\) be a sequence of independent identically distributed random variables with the distribution function \(F(x)\) and \(M\xi_{i}=0,\;D\xi_{i}=1,\;i=1,2,\ldots\). Let \(F_{n}(x)=P\{(\xi_1+\ldots +\xi_{n})/\sqrt{n}<x\}\), let \(\Phi(x)\) be the standard normal law distribution function, and let \(\nu_3=\int_{-\infty}^{+\infty}|x^3||d(F(x)-\Phi(x))|\). Then NEWLINE\[NEWLINE|F_{n}(x)-\Phi(x)|\leq C{\displaystyle \max(\nu_3,\nu_3^{1/6})\over (1+|x|)^2 \sqrt{n}}NEWLINE\]NEWLINE for all \(n>3\), where \(C\) is a constant.
scientific article; zbMATH DE number 1631417