A comparison theorem on moment inequalities between negatively associated and independent random variables
A comparison theorem on moment inequalities between negatively associated and independent random variables
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DOI：
10.1023/a:1007849609234
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发表时间：
2000-04-01
影响因子：
0.8
通讯作者：
Shao, QM
中科院分区：
文献类型：
--
作者：
Shao, QM
设{X-i， 1小于或等于i小于或等于n}为负相关序列，设{X-i*， 1小于或等于i小于或等于n}为独立随机变量序列，使得X-i*和X-i在每个i = 1,2，…，n时具有相同的分布。本文证明了对于R-1上的任何凸函数f， Ef(Sigma（i=1)(n) X-i）小于或等于Ef(Sigma(i=1)(n) X-i(*)), Ef(max(1小于或等于k小于或等于n) Sigma（i=k)(n) X-i）小于或等于Ef(max(1小于或等于k小于或等于n) Sigma（i=1)X(i)*）对于任何递增的凸函数。因此，大多数著名的不等式，如Rosenthal极大不等式和Kolmogorov指数不等式，对于负相关的随机变量仍然成立。特别地，关于负相关独立随机变量之间矩不等式的比较定理推广了Hoeffding不等式关于不替换有限总体的随机样本和的概率界。
Let {X-i, 1 less than or equal to i less than or equal to n} be a negatively associated sequence, and let {X-i*, 1 less than or equal to i less than or equal to n} be a sequence of independent random variables such that X-i* and X-i have the same distribution for each i = 1,2,...,n. It is shown in this paper that Ef(Sigma(i=1)(n) X-i) less than or equal to Ef(Sigma(i=1)(n) X-i(*)) for any convex function f on R-1 and that Ef(max(1 less than or equal to k less than or equal to n) Sigma(i=k)(n) X-i) less than or equal to Ef(max(1 less than or equal to k less than or equal to n) Sigma(i=1)X(i)*) for any increasing convex function. Hence, most of the well-known inequalities, such as the Rosenthal maximal inequality and the Kolmogorov exponential inequality, remain true For negatively associated random variables. In particular, the comparison theorem on moment inequalities between negatively associated iind independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population.