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{\heiti\bf ����1}\hy ��$\ep X=0$, ��$\ep X^2I_{(|X|\leq x)}$������Զ���ǻ��亯��, ��$$
\lim_{\varepsilon\searrow0}\varepsilon^2\tsm_{n\geq3}\frac{1}{n\log n}
\pr\Big(\Big|\frac{S_n}{V_n}\Big|\geq\varepsilon\sqrt{\log\log n}\Big)=1.
$${\heiti\bf ����2}\hy ��$\ep X=0$, ��$\ep X^2I_{(|X|\leq x)}$������Զ���ǻ��亯��, ���$0\leq\delta\leq1$, ��$$
\lim_{\varepsilon\searrow0}\varepsilon^{2\delta+2}\tsm_{n\geq1}
\frac{(\log n)^{\delta}}{n}\pr\Big(\Big|\frac{S_n}{V_n}\Big|
\geq\varepsilon\sqrt{\log n}\Big)=\frac{1}{\delta+1}\ep|N|^{2\delta+2},
$$
����$N$Ϊ��׼��̬���������

In this paper we obtained the precise asymptotics in Davis's law of law numbers and LIL for self-normalized sums, i.e.{\bf Theorem 1}\hy Let $\ep X=0$, and $\ep X^2I_{(|X|\leq x)}$ is slowly varying at $\infty$, then
$$\lim_{\varepsilon\searrow0}\varepsilon^2\tsm_{n\geq3}\frac{1}{n\log n}
\pr\Big(\Big|\frac{S_n}{V_n}\Big|\geq\varepsilon\sqrt{\log\log n}\Big)=1.
$${\bf Theorem 2}\hy Let $\ep X=0$, and $\ep X^2I_{(|X|\leq x)}$ is slowly varying at $\infty$, then for $0\leq\delta\leq1$, we have
$$ \lim_{\varepsilon\searrow0}\varepsilon^{2\delta+2}\tsm_{n\geq1}
\frac{(\log n)^{\delta}}{n}\pr\Big(\Big|\frac{S_n}{V_n}\Big|
\geq\varepsilon\sqrt{\log n}\Big)=\frac{1}{\delta+1}\ep|N|^{2\delta+2},
$$ where $N$ denote the standard normal random variable.