Comparison of threshold stop rules and maximum for independent nonnegative random variables. (English) Zbl 0549.60036

Let \(X_ 1,...,X_ n\) be a sequence of independent random variables, and let \(T_ n\) be the set of stopping rules for \(X_ 1,...,X_ n\). U. Krengel and L. Sucheston [On semiamarts, amarts, and processes with finite value, in J. Kuelbs (ed.), Probability on Banach spaces. Adv. Probab. Related Top. Vol. 4 (1978; Zbl 0394.62002), 197-266] have shown that (1) \(E(\max\{X_ 1,...,X_ n\})\leq 2 \sup\{EX_ t,\quad t\in T_ n\}\). Moreover, R. P. Kertz [Stop rule and supremum expectations of i.i.d. random variables: A complete comparison by conjugate duality. Unpublished manuscript (1983)] has shown that in the case of i.i.d. random variables, the constant 2 can be replaced by \(1+\alpha^*=1.341...\), where \(\alpha^*\) is the unique solution to \(\int^{1}_{0}(y-y \ln y+\alpha)^{-1}dy=1.\)
It is shown that inequality (1) still holds when the set of stopping rules \(T_ n\) is replaced by the set of rules in the form of \(t(c)=\min\{i<n:X_ i\geq c\}\) and \(t(c)=n\) if no such i exists. However, the constant 2 cannot be improved upon (for large n), when considering this subset of rules, even when the \(X_ i's\) are i.i.d.
Reviewer: J.Gianini-Pettitt


60G40 Stopping times; optimal stopping problems; gambling theory


Zbl 0394.62002
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