Some best possible prophet inequalities for convex functions of sums of independent variates and unordered martingale difference sequences

Abstract

Let Φ(·) be a nondecreasing convex function on [0, ∞). We show that for any integer n ≥ 1 and real a, EΦ((Mn - a)+) ≤ 2EΦ((Sn - a)+) - Φ(0) and E(Mn ∨ med Sn) ≤ E|Sn -med Sn|. where X1, X2, . . . are any independent mean zero random variables with partial sums S0 = 0, Sk = X1 + . . . + Xk and partial sum maxima Mn = max0≤k≤nSk. There are various instances in which these inequalities are best possible for fixed n and/or as n → ∞. These inequalities remain valid if {Xk} is a martingale difference sequence such that E(Xk | {Xi: i ≠ k}) = 0 a.s. for each k ≥ 1. Modified versions of these inequalities hold if the variates have arbitrary means but are independent.