Maximizing \(E\max _{1\leq k\leq n}S^+_ k/ES^+_ n:\) A prophet inequality for sums of i.i.d. mean zero variates. (English) Zbl 0684.60032

Klass, Michael J.

Ann. Probab. 17, No. 3, 1243-1247 (1989).

Let \(X_ 1,X_ 2,..\). be i.i.d. mean zero random variables. Put \(S_ k=X_ 1+...+X_ k\). It is proved that for any \(n\geq 1\) \[ E(\max_{1\leq k\leq n\quad}S^+_ k)\leq (2-n^{-1})E S\quad^+_ n. \] This result is nearly sharp.

Reviewer: U.Krengel

MSC:

60G40	Stopping times; optimal stopping problems; gambling theory
60E15	Inequalities; stochastic orderings
60G50	Sums of independent random variables; random walks

Keywords:

optimal stopping; prophet inequality