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CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST 2012, 28(3) 277-284 DOI: ISSN: 1001-4268 CN: 31-1256

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A Remark on the Tail Probability Sums of i.i.d. Gaussian Random Variable

He Jianjun,Xie Tingfan

Department of Mathematics, China Jiliang University

Abstract��

Let $\100dpi \inline $\{X,X_{n},n\geq1\}$$ be a sequence
of i.i.d. Gaussian random variables with zero mean and finite
variance, and set $\100dpi \inline $S_{n}=\tsm_{k=1}^{n}X_{k}$$ ,
$\100dpi \inline $\lambda(\epsilon)=\tsm_{n=1}^{\infty}\pr(|S_{n}|\geq n\epsilon)$$ .
In this paper, we prove that there exists positive constants $\100dpi \inline $C_{1}$$ and $\100dpi \inline $C_{2}$$ ,
for small enough $\epsilon>0$, it follows that
$\100dpi \inline $\epsilon>0$$ , it follows that
$\100dpi \inline $C_{1}\epsilon^{3}\leq$$ $\100dpi \inline $\epsilon^{2}\lambda(\epsilon)-\sigma^{2}+{\epsilon^{2}}/{2}\leq C_{2}\epsilon^{3}$$ .

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