LAWS OF LARGE NUMBERS FOR UNCORRELATED CES ` ARO UNIFORMLY INTEGRABLE RANDOM VARIABLES
LAWS OF LARGE NUMBERS FOR UNCORRELATED CES ` ARO UNIFORMLY INTEGRABLE RANDOM VARIABLES
复制标题
DOI：
--
复制
发表时间：
1997
期刊：
影响因子：
--
通讯作者：
D. Landers;L. Rogge
中科院分区：
文献类型：
--
作者：
D. Landers;L. Rogge
总结。证明了对于非负不相关随机变量，Cesaro-一致可积性包含弱大数定律，强Cesaro-一致可积性包含强大数定律。一个反例表明，在这种情况下，非否定假设是必不可少的。在Landers-Rogge（1987）中，证明了对独立一致可积随机变量的弱大数定律。Chandra（1989）将一致可积性假设弱化为Cesaro一致可积性假设，得到了成对独立随机变量的偶l1收敛性。我们证明钱德拉的结果适用于非负和不相关的随机变量，而不是两两独立的随机变量，但不适用于一般的不相关随机变量（见定理2(i)和例4）。Landers和Rogge（1987）进一步证明了一对独立和强一致可积随机变量的强大数定律（SLLN）。Chandra和Goswami（1992）证明了对独立和Cesaro强一致可积随机变量的更一般的SLLN。我们表明，他们的结果适用于非负的和不相关的，而不是两两独立的随机变量，但不是没有非负性的假设（见定理2（ii）和例4）。
SUMMARY. It is proved for non-negative uncorrelated random variables that Cesaro- uniform integrability implies the weak law of large numbers and strong Cesaro-uniform in- tegrability implies the strong law of large numbers. A counterexample shows that the non- negativity assumption is essential in this context. In Landers-Rogge (1987) a weak law of large numbers (WLLN) was proved for pairwise independent uniformly integrable random variables. Chandra (1989) weakened the assumption of uniform integrability to Cesaro uniform integrability and obtained even L1-convergence for pairwise independent random variables. We show that Chandra's result holds for non-negative and uncorrelated instead of pairwise independent random variables, but not for uncorrelated random vari- ables in general (see Theorem 2(i) and Example 4). Landers and Rogge (1987) furthermore proved a strong law of large num- bers (SLLN) for pairwise independent and strongly uniformly integrable random variables. Chandra and Goswami (1992) proved a more general SLLN for pair- wise independent and Cesaro strongly uniformly integrable random variables. We show that their result holds for non-negative and uncorrelated instead of pairwise independent random variables, but not without the assumption of non- negativity (see Theorem 2(ii) and Example 4).