Precise Asymptotics in Davis's Law of Large Numbers and the Iterated Logarithm for Self-Normalized Sums

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\frac1n\log n\pr\Big(\Big|\fracS_nV_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)^\deltan\pr\Big(\Big|\fracS_nV_n\Big|\geq\varepsilon\sqrt\log n\Big)=\frac1\delta+1\ep|N|^2\delta+2, where N denote the standard normal random variable.