论函数序列的期望值。
On the Expected Values of Sequences of Functions.
作者信息
Glueck Deborah H, Muller Keith E
机构信息
Dept. of Prev. Med. and Biometrics, University of Colorado HSC, Campus Box B119, 4200 East Ninth Avenue, Denver, CO 80262.
Dept. of Biostatistics, University of North Carolina, Chapel Hill, NC 27599-7400.
出版信息
Commun Stat Theory Methods. 2001 Jan 1;30(2). doi: 10.1081/STA-100002037.
We prove new extensions to lemmas about combinations of convergent sequences of distribution functions and absolutely continuous bounded functions. New lemma one, a generalized Helly theorem, allows computing the limit of the expected value of a sequence of functions with respect to a sequence of measures. Previously published results allow either the function or the measure to be a sequence, but not both. Lemma two allows computing the expected value of an absolutely continuous monotone function by integrating the probabilities of the inverse function values. Previous results were restricted to the identity function. Lemma three gives a computationally and analytically convenient form for the limit of the expected value of a sequence of functions of a sequence of random variables. This is a new result that follows directly from the first two lemmas. Although the lemmas resemble standard results and seem obviously true, we have found only similar looking and related but quite distinct results in the literature. We provide examples which highlight the value of the new results.
我们证明了关于分布函数收敛序列与绝对连续有界函数组合的引理的新扩展。新引理一，即广义赫利定理，允许计算关于一列测度的函数序列的期望值的极限。先前发表的结果允许函数或测度为序列，但不能两者皆为序列。引理二允许通过对反函数值的概率进行积分来计算绝对连续单调函数的期望值。先前的结果仅限于恒等函数。引理三给出了一列随机变量的函数序列的期望值极限的一种计算和分析上都很方便的形式。这是一个直接由前两个引理得出的新结果。尽管这些引理类似于标准结果且看似显然正确，但我们在文献中仅找到了外观相似且相关但截然不同的结果。我们提供了一些例子来突出这些新结果的价值。
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