Maximizing $E \max_{1 \leq k \leq n} S^+_k/ES^+_n$: A Prophet Inequality for Sums of I.I.D. Mean Zero Variates

Abstract

Let $X, X_1, X_2, \ldots$ be i.i.d. mean zero random variables. Put $S_k = X_1 + \cdots + X_k$. We prove that for every $n \geq 1, E \max_{1 \leq k \leq n} S^+_n \leq (2 - n^{-1})ES^+_n$. This result is nearly sharp, since if $P(X = 1) = P(X = -1) = \frac{1}{2},$ then $E \max{1 \leq k \leq n} S^+_k = (2 - n^{-1/2}\gamma^+_n)ES^+_n,$ where $\lim_{n \rightarrow \infty} \gamma^+_n = \sqrt{\pi/2}$.