On the accuracy of the normal approximation for sums of independent random variables
Let independent random variables \(X_1,\dotsc, X_n\) be defined on a probability space \((\Omega,{\mathcal A},\operatorname{P})\). Suppose that \(\operatorname{E} X_k= 0\), \(\sigma^2_k=\operatorname{E} X^2_k<\infty\), \(\beta_{2+\delta,k}= \operatorname{E}|X_k|^{2+\delta}<\infty\) for some \(\delta> 0\) and for all \(k= 1,\dotsc, n\). Denote \[ \begin{aligned} B^2_n &= \sum^n_{k=1} \sigma^2_k,\;l_n= B^{-(2+\delta)} \sum^n_{k=1} \beta_{2+\delta,k}, \beta_{\delta,k}= \operatorname{E}|X_k|^\delta,\\ \Delta_n &= \sup_x \Biggl|\operatorname{P}\{X_1+\dotsb+ X_n< xB_n\}- {1\over\sqrt{2\pi}} \int^x_{-\infty} e^{-u^2/2} du\Biggr|.\end{aligned} \] Recall the Prawitz's estimate \[ \Delta_n\leq {2\over 3\sqrt{2\pi}} {\beta_3\over \sigma^3\sqrt{n-1}}+ {1\over 2\sqrt{2\pi(n-1)}}+ A_1 l^2_{n-1},\;n\geq 2,\tag{1} \] for identically distributed random variables \(X_1,\dotsc, X_n\) with \(\sigma^2= \sigma^2_k\); the Bentkus' estimate \[ \Delta_n\leq {2l_n\over 3\sqrt{2\pi}}+ {1\over 2\sqrt{2\pi} B^3_n} \sum^n_{k=1} \sigma^3_k+ A_2 l^{4/3};\tag{2} \] the Chistyakov's estimate \[ \Delta_n\leq {\sqrt{10}+ 3\over 6\sqrt{2\pi}} l_n+ A_3 l^{40/39}_n|\ln l_n|^{7/6},\tag{3} \] where \(JA_1\), \(A_2\), \(A_3\) are some constants. The author of the present paper improves in some sense the estimates (1)--(3). Here is the main result of the paper. Theorem: For any \(0<\delta\leq 1\) and \(l> 0\) for all \(n\geq 1\) such that \(l_n\leq l\), we have the estimate \[ \Delta_n\leq C(\delta) l_n+ {1\over 2\sqrt{2\pi}} \Biggl(l_n+{1\over B^{2+\delta}_n} \beta_{\delta,k} \sigma^2_j\Biggr)^{1/5}+ A_n, \] where \(A_n=\overline C_\delta(l) l^{4/(2+\delta)}_n\) if \(\delta< 1\), and \(A_n=\overline C_1(l) l^{5/3}_n\) if \(\delta= 1\), \(\overline C_\delta(l)\) and \(\overline C_1(l)\) are some bounded nondecreasing functions, \[ C(\delta)= {\gamma_\delta 2^\delta\Gamma(1+ \delta/2)\over\pi}\, \gamma_\delta= \sup_{x>0} x^{-(2+\delta)}|e^{ix}- 1- ix-\textstyle{{1\over 2}}(ix)^2|,\;i= \sqrt{-1}. \] The author clarifies the theorem in some special cases.
scientific article; zbMATH DE number 6106440