Abstract
Let $\Phi (\cdot)$ be a nondecreasing convex function on $[0, \infty)$. We show that for any integer $n \geq 1$ and real $a$, $$E \Phi ((M_n - a)^+) \leq 2E \Phi ((S_n - a)^+) - \Phi (0)$$ and $$E(M_n \vee \mathrm{med} S_n) \leq E|S_n - \mathrm{med} S_n|.$$ where $X_1, X_2, \dots$ are any independent mean zero random variables with partial sums $S_0 = 0, S_k = X_1 + \dots + X_k$ and partial sum maxima $M_n = \max_{0 \leq k \leq n} S_k$. There are various instances in which these inequalities are best possible for fixed $n$ and/or as $n \to \infty$. These inequalities remain valid if $\{X_k\}$ is a martingale difference sequence such that $E(X_k \{X_i; i \not= k\}) = 0$ a.s. for each $k \geq 1$. Modified versions of these inequalities hold if the variates have arbitrary means but are independent.
Citation
K. P. Choi. Michael J. Klass. "Some best possible prophet inequalities for convex functions of sums of independent variates and unordered martingale difference sequences." Ann. Probab. 25 (2) 803 - 811, April 1997. https://doi.org/10.1214/aop/1024404420
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