A prophet inequality for -bounded dependent random variables

Let X = (X-n)(n >= 1) be a sequence of arbitrarily dependent nonnegative random variables satisfying the boundedness condition sup(tau)EX(tau)(p) <= t, where t > 0,1 < p < infinity are fixed numbers and the supremum is taken over all finite stopping times of X. Let M = E sup(n) X-n and V = sup(tau) EX tau denote the expected supremum and the optimal expected return of the sequence X, respectively. We establish the prophet inequality M <= V + V/p - 1 log (te/V-p) and show that the bound on the right is the best possible. The proof of the inequality rests on Burkholder's method and exploits properties of certain special functions. The proof of the sharpness is somewhat indirect, but we also provide an indication how the extremal sequences can be constructed.