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Precise asymptotics for a series of T. L. Lai

Authors:	Aurel Spataru

Institution:	Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Calea 13 Septembrie.13, 76100 Bucharest, Romania

Abstract:	Let $X,~X_{1},\,X_{2},...$ be i.i.d. random variables with , and set $S_{n}=X_{1}+...+X_{n}$ . We prove that, for $\begin{displaymath}\lim_{\varepsilon \searrow \sigma \sqrt{2p-2}}\sqrt{\varepsil... ...\geq \varepsilon \sqrt{n\log n} )=\sigma \sqrt{\frac{2}{p-1}}, \end{displaymath}$ under the assumption that $EX^{2}=\sigma ^{2}$ and $E\left\vert X\right\vert ^{2p}(\log ^{+}\left\vert X\right\vert )^{-p}]<\infty .$ Necessary and sufficient conditions for the convergence of the sum above were established by Lai (1974).

Keywords:	Tail probabilities of sums of i i d random variables moderate deviations Lai law

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