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{heitibf 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$lim_{varepsilonsearrow0}varepsilon^2tsm_{ngeq3}frac{1}{nlog n}prBig(Big|frac{S_n}{V_n}Big|geqvarepsilonsqrt{loglog n}Big)=1.${heitibf 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$$lim_{varepsilonsearrow0}varepsilon^2tsm_{ngeq3}frac{1}{nlog n}prBig(Big|frac{S_n}{V_n}Big|geqvarepsilonsqrt{loglog n}Big)=1.$${bf Theorem 2}hy Let $ep X=0$, and $ep X^2I_{(|X|leq x)}$ is slowly varying at $infty$, then for $0leqdeltaleq1$, we have$$ lim_{varepsilonsearrow0}varepsilon^{2delta+2}tsm_{ngeq1}frac{(log n)^{delta}}{n}prBig(Big|frac{S_n}{V_n}Big|geqvarepsilonsqrt{log n}Big)=frac{1}{delta+1}ep|N|^{2delta+2},$$ where $N$ denote the standard normal random variable.