Some results on convergence rates for probabilities of moderate deviations for sums of random variables
Summary: Let \(X\), \(X_ n\), \(n\geq 1\), be a sequence of i.i.d. real random variables, and \(S_ n=\sum^ n_{k=1}X_ k\), \(n\geq 1\). Convergence rates of moderate deviations are derived, i.e., the rate of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of series \(\sum_{n\geq 1}(\psi^ 2(n)/n)P(| S_ n|\geq\sqrt n\varphi(n))\) only under the assumptions that \(EX=0\) and \(EX^ 2=1\), where \(\varphi\) and \(\psi\) are taken from a broad class of functions. These results generalize and improve some recent results of \textit{D. Li} [Sci. China, Ser. A 34, No. 4, 395--404 (1991; Zbl 0735.60031)] and \textit{M. Gafurov} [Probability theory and mathematical statistics, Proc. 4th USSR Jap. Symp., Tbilisi/USSR 1982, Lect. Notes Math. 1021, 137--144 (1983; Zbl 0533.60039)] and some previous work of \textit{J. A. Davis} [Ann. Math. Stat. 39, 1479--1485 (1968; Zbl 0174.49902)]. For \(b\in[0,1]\) and \(\varepsilon>0\), let \[ \lambda_{\varepsilon,b}=\sum_{n\geq 3}((\log\log n)^ b/n)I(| S_ n|\geq\sqrt{(2+\varepsilon)n\log\log n}). \] The behaviour of \(E\lambda_{\varepsilon,b}\) as \(\varepsilon\downarrow 0\) is also studied.
scientific article; zbMATH DE number 89242