Prophet inequalities for I.I.D. random variables with random arrival times

Suppose X1, X2, . . . are i.i.d. nonnegative random variables with finite expectation, and for each k, Xk is observed at the k-th arrival time Sk of a Poisson process with unit rate which is independent of the sequence {Xk}. For t > 0, comparisons are made between the expected maximum M(t) := E[maxk≥1 Xk I(Sk ≤ t)] and the optimal stopping value V (t) := supτ∈T E[Xτ I(Sτ ≤ t)], where T is the set of all IN-valued random variables τ such that {τ = i} is measurable with respect to the σ-algebra generated by (X1, S1), . . . , (Xi, Si). For instance, it is shown that M(t)/V (t) ≤ 1 + α0, where α0 . = 0.34149 satisfies ∫ 1 0 (y − y ln y + α0) −1 dy = 1; and this bound is asymptotically sharp as t → ∞. Another result is that M(t)/V (t) < 2 − (1 − e)/t, and this bound is asymptotically sharp as t ↓ 0. Upper bounds for the difference M(t)−V (t) are also given, under the additional assumption that the Xk are bounded. AMS 2000 subject classification: 60G40, 62L15.