## 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 Unpublished manuscript (1983)] has shown that in the case of i.i.d. random variables, the constant 2 can be replaced by $$1+\alpha^*=1.341...$$, where $$\alpha^*$$ is the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

### MSC:

 60G40 Stopping times; optimal stopping problems; gambling theory

### Keywords:

prophet inequalities; stopping rules

Zbl 0394.62002
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