摘要
{X,Xi,i≥1}是i.i.d.r.v′.s.在矩母函数存在的条件下,由古典的Erdos-Rényi大数律有limn→∞max0≤k≤n∑k+[clogn]i=k+1Xi[clogn]=α(c),α(c)为某常数.自正则下MiklósCsorgo&ShaoQiman(1994)在仅要求一阶矩的条件下就得到了:limn→∞max0≤k≤n∑k+[clogn]i=k+1Xi∑k+[clogn]i=k+1(X2i+1)=β(c),β(c)为某常数.众所周知,自正则下人们往往在较弱条件下取得相应结果是因为:分母中的X能有效抵销分子中X较大而引起整个分式极限行为的波动.因此,在什么样的条件下,式max0≤k≤n∑k+[clogn]i=k+1Xi∑k+[clogn]i=k+1X2i1-β[clogn]β→r(c)成为非常有意思的问题,因为它将依赖于β的大小.本文给出,当0<β≤12时,只要E(X)≥0,上式就有有限极限.当12<β<1时,则必须在矩母函数存在下,上式才有有限极限.并都求出了其极限表达式.
X,X i,i≥1} is i.i.d.r.v′.s, It is well known that lim n→∞ max 0≤k≤n∑k+[c log n]i=k+1X i[c log n]=α(c),(α(c) is a constant depending on c), when E(e t 0X )<∞, t 0>0. By normalizing, paper got lim n→∞ max 0≤k≤n∑k+[c log n]i=k+1X i∑k+[c log n]i=k+1(X 2 i+1)=β(c),(β(c) is a constant) only under the condition of one moment. In this paper it is studied that under what conditions one can gets lim n→∞ max 0≤k≤n∑k+[c log n]i=k+1X i∑k+[c log n]i=k+1X 2 i 1-β [c log n] β→r(c), (r(c) is a constant) when β varies from 0 to 1, the problem has been solved completely.
出处
《高校应用数学学报(A辑)》
CSCD
北大核心
1997年第4期405-416,共12页
Applied Mathematics A Journal of Chinese Universities(Ser.A)
基金
国家和浙江省自然科学基金
关键词
自正则
E-R大数定律
大数定律
Erds Rényi Law of Large Numbers, Self normalization.