Precise asymptotics for a series of T. L. Lai
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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{equation*} \lim _{\varepsilon \searrow \sigma \sqrt {2p-2}}\sqrt {\varepsilon ^{2}-\sigma ^{2}(2p-2)}\sum _{n\geq 2}n^{p-2}P(|S_{n}|\geq \varepsilon \sqrt {n\log n} )=\sigma \sqrt {\frac {2}{p-1}}, \end{equation*} 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).References
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Additional Information
- Aurel Spătaru
- Affiliation: Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Calea 13 Septembrie.13, 76100 Bucharest, Romania
- Email: aspataru@pcnet.ro
- Received by editor(s): August 1, 2003
- Published electronically: June 21, 2004
- Communicated by: Richard C. Bradley
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2073316