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Journal Article

# Comparisons of Stop Rule and Supremum Expectations of I.I.D. Random Variables

T. P. Hill and Robert P. Kertz
The Annals of Probability
Vol. 10, No. 2 (May, 1982), pp. 336-345
Stable URL: http://www.jstor.org/stable/2243434
Page Count: 10

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Topics: Random variables, Mathematical constants, Gambling, Games, Prophets

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## Abstract

Implicitly defined (and easily approximated) universal constants $1.1 < a_n < 1.6, n = 2,3, \cdots$, are found so that if X1, X2, ⋯ are i.i.d. non-negative random variables and if Tn is the set of stop rules for X1, ⋯, Xn, then $E(\max\{X_1, \cdots, X_n\}) \leq a_n \sup\{EX_t: t \in T_n\}$, and the bound an is best possible. Similar universal constants $0 < b_n < \frac{1}{4}$ are found so that if the {Xi} are i.i.d. random variables taking values only in [ a, b], then $E(\max\{X_1, \cdots, X_n\}) \leq \sup\{EX_t: t \in T_n\} + b_n(b - a)$, 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.

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