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- W2953037548 abstract "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 $M(t):=rE[max_{kgeq 1} X_k sI(S_kleq t)]$ and the optimal stopping value $V(t):=sup_{tauinTT}sE[X_tau sI(S_tauleq t)]$, 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 $alpha_0doteq 0.34149$ satisfies $int_0^1(y-yln y+alpha_0)^{-1} dy=1$; and this bound is asymptotically sharp as $ttoinfty$. Another result is that $M(t)/V(t)<2-(1-e^{-t})/t$, and this bound is asymptotically sharp as $tdownarrow 0$. Upper bounds for the difference $M(t)-V(t)$ are also given, under the additional assumption that the $X_k$ are bounded." @default.
- W2953037548 created "2019-06-27" @default.
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- W2953037548 date "2006-11-21" @default.
- W2953037548 modified "2023-09-27" @default.
- W2953037548 title "Prophet inequalities for i.i.d. random variables with random arrival times" @default.
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