On the accuracy of the normal approximation for sums of symmetric independent random variables
Let \(X_1, X_2, \dotsc , X_n\) be independent symmetric random variables defined on a probability space \((\Omega,{\mathcal A}, \operatorname{P})\). Suppose \(\operatorname{E}|X_j|^{2+\delta} < +\infty\), \(j=1,2,\dotsc , n\), for some \(\delta \in [0,1]\). Write \[ \sigma_j^2 = \operatorname{E}(X_j^2), \quad \beta_{2+\delta,j} = \operatorname{E}(|X_j|^{2+\delta}),\quad \beta_{\delta,j} = \operatorname{E}(|X_j|^{\delta}),\quad B_n^2 = \sum_{j=1}^n \sigma_j^2, \] \[ l_n = \frac{1}{B_n^{2+\delta}} \sum_{j=1}^n \beta_{2+\delta,j}, \quad \Delta_n = \sup_{x} \left| \overline{F}_n(x) - \Phi(x)\right|, \] where \(\overline{F}_n(x) = \operatorname{E}\left(X_1 + X_2 + \dotsb + X_n < x B_n\right)\), and \(\Phi \) is the standard normal distribution function. This paper investigates bounds for the Kolmogorov metric \(\Delta_n\) in terms of the above quantities. The results presented are related to the results obtained by the author without the assumption of symmetry [Dokl. Math. 85, No. 2, 274--278 (2012); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 443, No. 5, 555--560 (2012; Zbl 1269.60028)]. Let \[ k_{\delta} = \sup_{x>0} \frac{\cos x -1 + \frac{x^2}{2}}{x^{2+\delta}}, \qquad 0 < \delta \leq 1. \] For the case \(\delta = 1\), \textit{V. Bentkus} asserted that \[ \Delta_n \leq \frac{1+4 k_1}{2\sqrt{2\pi}}\, l_n + \frac{1-4 k_1}{2\sqrt{2\pi}}\, \tau_n + A \cdot l_n^{4/3}, \qquad \text{where } \tau_n = \frac{1}{B_n^3}\, \sum_{j=1}^n \sigma_j^3, \] and \textit{G. P. Chistyakov} proved \[ \Delta_n \leq \frac{l_n}{\sqrt{2\pi}} + B\cdot l_n^{\frac{40}{39}}\, |\ln l_n|^{7/6}, \] where \(A\) and \(B\) are absolute constants [Theory Probab. Appl. 46, No. 2, 226--242 (2001); translation from Teor. Veroyatn. Primen. 46, No. 2, 326--344 (2001; Zbl 1008.60034); Theory Probab. Appl. 46, No. 3, 516--522 (2001); translation from Teor. Veroyatn. Primen. 46, No. 3, 573--578 (2001; Zbl 1034.60020)]. In this paper, the author proposes several new bounds for \(\Delta_n\), which result in an improvement of Chistyakov's estimate and in some cases of Bentkus'. In particular, for any \(l>0\), for all \(n\geq 1\) and \(X_1, X_2, \dotsc, X_n\) such that \(l_n \leq l\), the author gives the following estimates. For \(0 < \delta \leq 1\), \[ \Delta_n \leq C_s(\delta) \cdot l_n + \frac{1}{2\, \sqrt{2 \pi}} \left( l_n + \frac{1}{B_n^{2+\delta}} \sum_{j=1}^n \beta_{\delta,j} \sigma_j^2 \right)^{1/\delta} + \tilde{C}_{\delta}(l) \cdot l_n^{4/(2+\delta)}, \] for \(\delta =1\), \[ \Delta_n \leq \frac{\frac{1}{2} +k_{1}}{\sqrt{2\pi}}\, l_n + \frac{\frac{1}{2} - k_{1} }{\sqrt{2\pi} B_n^3}\, \sum_{j=1}^n \beta_{1,j} \sigma_j^2 + \begin{cases} \tilde{C}(l) \cdot l_n^{7/6} & \text{ in the general case,}\\ \hat{C}(l)\cdot l_n^{3/2} & \text{ if } X_1 \overset{d}{=}X_2\overset{d}{=}\dotsb \overset{d}{=} X_n. \end{cases} \] As a consequence, \[ \Delta_n \leq \frac{1}{\sqrt{2\pi}}\, l_n + 3.4418\, l_n^{7/6} \quad \text{for any } l_n, \] where \[ C_s(\delta) = k_{\delta}\, 2^{\delta/2}\,\frac{\Gamma (1+\delta/2)}{\pi}, \] and \(\tilde{C}_{\delta}(l), \tilde{C}(l), \hat{C}(l) \) are bounded monotonically nondecreasing functions. Some values for the majorants of these constants are given.
scientific article; zbMATH DE number 6106445