Comparisons of Stop Rule and Supremum Expectations of I.I.D. Random Variables
Comparisons of Stop Rule and Supremum Expectations of I.I.D. Random Variables
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DOI：
10.1214/aop/1176993861
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发表时间：
1982-05
影响因子：
2.3
通讯作者：
T. Hill;R. P. Kertz
中科院分区：
文献类型：
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作者：
T. Hill;R. P. Kertz
隐式定义（且易于逼近）的普适常数1.1 < an 随机变量，并且如果Tn是针对X1，…，Xn的停时规则集合，那么E(max{X1，…，Xn}) ≈ an sup {EXt : t∈Tn}，并且该界an是最优的。找到了类似的普适常数0 < bn < Y，使得如果{Xi}是独立同分布随机变量且仅取值于[a, b)，那么E(max{X1，…，Xn}) ≈ sup {EXt : t∈Tn} + bn(b - a)，这里界bn同样是最优的。在这两种情形下，达到（或近乎达到）等式的极值分布以隐式形式给出。
Implicitly defined (and easily approximated) universal constants 1.1 < an random variables and if Tn is the set of stop rules for Xl, "', Xn, then E(max{Xl , • • • ,Xn}) ~ an sup {EX, : tE Tn}, and the bound an is best possible. Similar universal constants 0 < bn < Y. are found so that if the {Xi} are i.i.d. random variables taking values only in [a, b), then E (max {XI , .. , , Xn }) ~ sup {EXt: t E Tn} + bn(b-a), where again the bound bn is best possible. In both situations, extremal distributions for which equality is attained (or nearly attained) are given in implicit form.