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

Keywords��