自正则化和Davis大数律和重对数律的精确渐近性

本文证明了自正则化Davis大数律和重对数律的精确渐近性, 即 {\heiti\bf 定理1}\hy 设$\ep X=0$, 且$\ep X^2I_{(|X|\leq x)}$在无穷远处是缓变函数, 则$\lim_{\varepsilon\searrow0}\varepsilon^2\tsm_{n\geq3}\frac{1}{n\log n}\pr\Big(\Big|\frac{S_n}{V_n}\Big|\geq\varepsilon\sqrt{\log\log n}\Big)=1.${\heiti\bf 定理2}\hy 设$\ep X=0$, 且$\ep X^2I_{(|X|\leq x)}$在无穷远处是缓变函数, 则对本文证明了目正则化Davis大数律和重对数律的精确渐近性,即定理1设EX=0,且EX~2I_(|x|≤x)在无穷远处是缓变函数,则■定理2设EX=0,且EX~2I_(|x|≤x)在无穷远处是缓变函数,则对0≤δ≤1,有■其中N为标准正态随机变量．

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$$\lim_{\varepsilon\searrow0}\varepsilon^2\tsm_{n\geq3}\frac{1}{n\log n}\pr\Big(\Big|\frac{S_n}{V_n}\Big|\geq\varepsilon\sqrt{\log\log n}\Big)=1.$${\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.