摘要
本学位论文是我在攻读硕士学位期间,在导师林正炎教授的悉心指导下完成的。全文共分三章,主要研究混合相依随机序列的极限性质。
第一章介绍了本论文的背景与本学科的一些发展概况,撰写本文的意义所在以及相依随机变量在投资组合中的应用。
第二章是本文的重点所在,讨论了同分布负坐标相依(NQD)随机序列的加权和的强大数律。设{X,X_n,n≥1}是一列均值为0的随机变量序列,而{a_(ni),1≤i≤n,n≥1}是常数组列,独立情形下加权和的完全收敛性质已经被许多学者研究过(如Choi和Sung,1987:Cuzick,1995;Wu,1999:Bai和Cheng,2000:Sung,2001等)。本文将证明类似的强大数律对于NQD随机变量也成立,在证明过程中我们利用了一个比较有效的方法使得证明过程较独立情形有所简化。同时,在Sung的结果中,对0≤r≤1情形要求b_n=n~(1/α)log~(1/r+β)n(β>0),我们取到b_n=n~(1/α)log~(1/r)n从而推广并实质地改进了Sung的结果。主要结果有:
定理2.2.1 令{X,X_n,n≥1}是一列均值为0的同分布的NQD序列,且存在β>1使得E|X|~β<∞。令{α_(ni),1≤i≤n,n≥1}是满足条件(2.2)(对某个α>1)的常数组列。则有这里1/p=1/α+1/β。相反地,如果对任意满足条件(2.2)的常数组列都有(2.4)式成立,则E|X|~β<∞且EX=0。
定理2.2.2 令{X,X_n,n≥1}是一列均值为0同分布的NQD随机序列,存在h>0,r>0使得E[exp(h|X|~r)]<∞。令{α_(ni),1≤i≤n,n≥1}是满足条件(2.2)(对某个
This article consists of three parts. The first part is concerned with the background of the subject, the application of dependent random variables and the significance of this article.In the second part, we show that two strong laws of large numbers for weighted sum, which were proved for arrays of i.i.d. random variables by Bai, Cheng (2000) and Sung (2001), can be also obtained for arrays of negative quadrant dependent (NQD) random variables. We use a more effective method to prove them, so that the proof is somewhat simplified. Moreover for Sung's result in the case of 0≤r≤1,wetake b_n= n~(1/a) log~(1/r) n instead of b_n= n~(1/a) log~((1/r)+β) n(β > 0). Thus, our result extendsand sharpens Sung's theorem.In the last part, we extend a result on complete convergence for the movingaverage process under independence assumptions to the case of ρ~- -mixing random variables sequences.
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