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Precise asymptotics for a series of T. L. Lai


Author: Aurel Spataru
Journal: Proc. Amer. Math. Soc. 132 (2004), 3387-3395
MSC (2000): Primary 60G50, 60E15
DOI: https://doi.org/10.1090/S0002-9939-04-07524-0
Published electronically: June 21, 2004
MathSciNet review: 2073316
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Abstract: Let $X,~X_{1},\,X_{2},...$ be i.i.d. random variables with $EX=0$, and set $S_{n}=X_{1}+...+X_{n}$. We prove that, for $1<p<3/2,$

\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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Additional Information

Aurel Spataru
Affiliation: Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Calea 13 Septembrie.13, 76100 Bucharest, Romania
Email: aspataru@pcnet.ro

DOI: https://doi.org/10.1090/S0002-9939-04-07524-0
Keywords: Tail probabilities of sums of i.i.d. random variables, moderate deviations, Lai law
Received by editor(s): August 1, 2003
Published electronically: June 21, 2004
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2004 American Mathematical Society

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