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ISSN 1088-6826(e) ISSN 0002-9939(p)

Precise asymptotics for a series of T. L. Lai

Author(s): Aurel Spataru
Journal: Proc. Amer. Math. Soc. 132 (2004), 3387-3395.
MSC (2000): Primary 60G50, 60E15
Posted: June 21, 2004
MathSciNet review: 2073316
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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).

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