随机变量组（序）列的收敛性和精确渐近性
摘要
本文第一章研究了具有很强的应用背景的混合鞅的最大值不等式和大数定律.第一章第二节中在不要求平方可积的条件下得到与McLeish(1975)类似的最大值不等式.作为推论,得到了混合鞅序列的强大数定律.第三节中在不要求一致可积的条件下得到与Andrews(1988)类似的弱大数定律.
第二章第二节中我们在比Cesaro一致可积弱的“一致Cesaro型”条件下研究了随机变量组列的弱收敛性,得到了任意随机变量组列的一类弱大数定律和两两NQD列的一类弱大数定律.作为推论,得到了Hong和Oh(1995)的结果.第三节中在不含有log n项的情况下,我们研究了两两NQD列的几乎必然收敛性,得到了两个主要结果.第一个是两两NQD列的强大数律.作为推论,我们得到了两两NQD列的Chung型强大数定律.第二个是得到了一个与独立序列情形类似的结果：若∑(?)<∞且supn≥1n-1∑(?)1E|Xk-EXk|=O(1),则limn→∞n-1(?)(Xk-EXk)=0 a.s.
第三章主要研究了(?)混合序列的收敛性.在第三章第二节我们讨论了(?)混合序列的强大数定律中部分和系数及权系数都为一般函数的情形,得到了两个一般性的强大数定律.只要取满足定理条件的不同的函数,便可得到一些具体形式的强大数定律.例如,我们得到了(?)混合序列的对数型强大数定律,Marcinkiewicz型强大数定律和Marcinkiewicz强大数定律.同时我们给出了(?)混合序列的Chung型强大数定律.另外我们得到了(?)混合序列的一类完全收敛性.第三节中我们在比Cesaro一致可积弱的“一致Cesaro型”条件下得到(?)混合序列的弱大数定律和Lr收敛性.
第四章主要研究了随机变量序列的精确渐近性,得到了独立同分布随机变量序列精确渐近性的一般性结果,也就是拟权函数和边界函数为一般函数时的情形,同时明显地揭示了拟权函数和边界函数之间的关系.作为推论,我们得到了Gut和Spataru(2000a,2000b)中的定理2和定理3.
第二章第二节中我们在比Cesaro一致可积弱的“一致Cesaro型”条件下研究了随机变量组列的弱收敛性,得到了任意随机变量组列的一类弱大数定律和两两NQD列的一类弱大数定律.作为推论,得到了Hong和Oh(1995)的结果.第三节中在不含有log n项的情况下,我们研究了两两NQD列的几乎必然收敛性,得到了两个主要结果.第一个是两两NQD列的强大数律.作为推论,我们得到了两两NQD列的Chung型强大数定律.第二个是得到了一个与独立序列情形类似的结果：若∑(?)<∞且supn≥1n-1∑(?)1E|Xk-EXk|=O(1),则limn→∞n-1(?)(Xk-EXk)=0 a.s.
第三章主要研究了(?)混合序列的收敛性.在第三章第二节我们讨论了(?)混合序列的强大数定律中部分和系数及权系数都为一般函数的情形,得到了两个一般性的强大数定律.只要取满足定理条件的不同的函数,便可得到一些具体形式的强大数定律.例如,我们得到了(?)混合序列的对数型强大数定律,Marcinkiewicz型强大数定律和Marcinkiewicz强大数定律.同时我们给出了(?)混合序列的Chung型强大数定律.另外我们得到了(?)混合序列的一类完全收敛性.第三节中我们在比Cesaro一致可积弱的“一致Cesaro型”条件下得到(?)混合序列的弱大数定律和Lr收敛性.
第四章主要研究了随机变量序列的精确渐近性,得到了独立同分布随机变量序列精确渐近性的一般性结果,也就是拟权函数和边界函数为一般函数时的情形,同时明显地揭示了拟权函数和边界函数之间的关系.作为推论,我们得到了Gut和Spataru(2000a,2000b)中的定理2和定理3.
引文
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