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

Implicitly defined (and easily approximated) universal constants 1. 1 < aₙ < 1. 6, n = 2, 3, , 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 (\X₁, , Xₙ\) aₙ \EXₜ: t Tₙ\, and the bound aₙ is best possible. Similar universal constants 0 < bₙ < 14 are found so that if the \Xᵢ\ are i. i. d. random variables taking values only in a, b, then E (\X₁, , Xₙ\) \EXₜ: t 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.