Prophet inequalities for i.i.d. random variables with random arrival times

Details Prophet inequalities for i.i.d. random variables with random arrival times Prophet inequalities for i.i.d. random variables with random arrival times

2006-11-21

arxiv.org/abs/math/0611664v1

Sequential Anal. 26, no. 4, 403-413 (2007)

Mathematics

Probability

16 pages with 1 figure; submitted to Sequential Analysis in shortened form

Scientific paper

Suppose $X_1,X_2,...$ are i.i.d. nonnegative random variables with finite expectation, and for each $k$, $X_k$ is observed at the $k$-th arrival time $S_k$ of a Poisson process with unit rate which is independent of the sequence $\{X_k\}$. For $t>0$, comparisons are made between the expected maximum $M(t):=\rE[\max_{k\geq 1} X_k \sI(S_k\leq t)]$ and the optimal stopping value $V(t):=\sup_{\tau\in\TT}\sE[X_\tau \sI(S_\tau\leq t)]$, where $\TT$ is the set of all $\NN$-valued random variables $\tau$ such that $\{\tau=i\}$ is measurable with respect to the $\sigma$-algebra generated by $(X_1,S_1),...,(X_i,S_i)$. For instance, it is shown that $M(t)/V(t)\leq 1+\alpha_0$, where $\alpha_0\doteq 0.34149$ satisfies $\int_0^1(y-y\ln y+\alpha_0)^{-1} dy=1$; and this bound is asymptotically sharp as $t\to\infty$. Another result is that $M(t)/V(t)<2-(1-e^{-t})/t$, and this bound is asymptotically sharp as $t\downarrow 0$. Upper bounds for the difference $M(t)-V(t)$ are also given, under the additional assumption that the $X_k$ are bounded.

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