The Annals of Probability

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

K. P. Choi and Michael J. Klass

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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 $$E(M_n \bigvee \med S_n) \leq E|S_n - \med S_n|.$$ 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.

Article information

Source
Ann. Probab., Volume 25, Number 2 (1997), 803-811.

Dates
First available in Project Euclid: 18 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1024404420

Digital Object Identifier
doi:10.1214/aop/1024404420

Mathematical Reviews number (MathSciNet)
MR1434127

Zentralblatt MATH identifier
0880.60017

Subjects
Primary: 60E15: Inequalities; stochastic orderings 60G50: Sums of independent random variables; random walks
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G42: Martingales with discrete parameter 60J15

Keywords
Maximum of partial sums sums of independent random variables prophet inequalities median unordered martingale difference sequence convex function

Citation

Choi, K. P.; Klass, Michael J. Some best possible prophet inequalities for convex functions of sums of independent variates and unordered martingale difference sequences. Ann. Probab. 25 (1997), no. 2, 803--811. doi:10.1214/aop/1024404420. https://projecteuclid.org/euclid.aop/1024404420


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References

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