Title

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

Author Info

Theodore P. Hill, Georgia Institute of Technology - Main CampusFollow
Robert P. Kertz, Georgia Institute of Technology - Main Campus

Recommended Citation

Published in The Annals of Probability, Volume 10, Issue 2, May 1, 1982, pages 336-345. Copyright © 1982 Institute of Mathematical Statistics. The definitive version is available at http://www.jstor.org/stable/2243434.

NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.

Abstract

Implicitly defined (and easily approximated) universal constants 1.1 < a_n < 1.6, n = 2,3, ... , are found so that if X₁, X₂, ... are i.i.d. non-negative random variables and if the T_n is the set of stop rules for X₁, ..., X_n, then E (max {X₁, ..., X_n}) ≤ a_nsup {EX_t : t ε T_n}, and the bound a_n is best possible. Similar universal constants 0 < b_n < 1/4 are found so that if the (X_i) are i.i.d. random variables taking values only in [a,b], then E (max {X₁, ..., X_n}) ≤ a_nsup {EX_t : t ε T_n} + b_n (b - a ), where again the bound b_n is the nest possible. In both situations, extremal distributions for which equality is attained (or nearly attained) are given in implicit form.

Disciplines

Mathematics