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April 1997 Some best possible prophet inequalities for convex functions of sums of independent variates and unordered martingale difference sequences
K. P. Choi, Michael J. Klass
Ann. Probab. 25(2): 803-811 (April 1997). DOI: 10.1214/aop/1024404420

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.

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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

Information

Published: April 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0880.60017
MathSciNet: MR1434127
Digital Object Identifier: 10.1214/aop/1024404420

Subjects:
Primary: 60E15, 60G50
Secondary: 60G40, 60G42, 60J15

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.25 • No. 2 • April 1997
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