Prophet Inequalities for I.I.D. Random Variables with Random Arrival Times

Abstract

Suppose X 1, X 2,… are independent and identically distributed nonnegative random variables with finite expectation, and for each k, X k is observed at the kth arrival time S k of a Poisson process with a unit rate independent of the sequence {X k }. For t > 0, comparisons are made between the expected maximum M(t) := E[max k≥1 X k I(S k ≤ t)] and the optimal stopping value V(t) := supτ∈𝒯E[X τ I(S τ ≤ t)], where 𝒯 is the set of all ℕ-valued random variables τ such that {τ = i} is measurable with respect to the σ-algebra generated by (X 1, S 1),…, (X i , S i ). For instance, it is shown that M(t)/V(t) ≤ 1 + α0, where α0 ≐ 0.34149 satisfies , and this bound is asymptotically sharp as t → ∞. Another result is that M(t)/V(t) < 2 − (1 − e −t )/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 X k 's are bounded.