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May, 1982 Comparisons of Stop Rule and Supremum Expectations of I.I.D. Random Variables
T. P. Hill, Robert P. Kertz
Ann. Probab. 10(2): 336-345 (May, 1982). DOI: 10.1214/aop/1176993861


Implicitly defined (and easily approximated) universal constants $1.1 < a_n < 1.6, n = 2,3, \cdots$, are found so that if $X_1, X_2, \cdots$ are i.i.d. non-negative random variables and if $T_n$ is the set of stop rules for $X_1, \cdots, X_n$, then $E(\max\{X_1, \cdots, X_n\}) \leq a_n \sup\{EX_t: t \in T_n\}$, and the bound $a_n$ is best possible. Similar universal constants $0 < b_n < \frac{1}{4}$ are found so that if the $\{X_i\}$ are i.i.d. random variables taking values only in $\lbrack a, b\rbrack$, then $E(\max\{X_1, \cdots, X_n\}) \leq \sup\{EX_t: t \in T_n\} + b_n(b - a)$, where again the bound $b_n$ is best possible. In both situations, extremal distributions for which equality is attained (or nearly attained) are given in implicit form.


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T. P. Hill. Robert P. Kertz. "Comparisons of Stop Rule and Supremum Expectations of I.I.D. Random Variables." Ann. Probab. 10 (2) 336 - 345, May, 1982.


Published: May, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0483.60035
MathSciNet: MR647508
Digital Object Identifier: 10.1214/aop/1176993861

Primary: 60G40
Secondary: 62L15 , 90C99

Keywords: extremal distributions , inequalities for stochastic processes , Optimal stopping

Rights: Copyright © 1982 Institute of Mathematical Statistics


Vol.10 • No. 2 • May, 1982
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