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Title: | Some best possible prophet inequalities for convex functions of sums of independent variates and unordered martingale difference sequences | Authors: | Choi, K.P. Klass, M.J. |
Keywords: | Convex function Maximum of partial sums Median Prophet inequalities Sums of independent random variables Unordered martingale difference sequence |
Issue Date: | Apr-1997 | Citation: | Choi, K.P.,Klass, M.J. (1997-04). Some best possible prophet inequalities for convex functions of sums of independent variates and unordered martingale difference sequences. Annals of Probability 25 (2) : 803-811. ScholarBank@NUS Repository. | Abstract: | Let Φ(·) be a nondecreasing convex function on [0, ∞). We show that for any integer n ≥ 1 and real a, EΦ((Mn - a)+) ≤ 2EΦ((Sn - a)+) - Φ(0) and E(Mn ∨ med Sn) ≤ E|Sn -med Sn|. where X1, X2, . . . are any independent mean zero random variables with partial sums S0 = 0, Sk = X1 + . . . + Xk and partial sum maxima Mn = max0≤k≤nSk. There are various instances in which these inequalities are best possible for fixed n and/or as n → ∞. These inequalities remain valid if {Xk} is a martingale difference sequence such that E(Xk | {Xi: i ≠ k}) = 0 a.s. for each k ≥ 1. Modified versions of these inequalities hold if the variates have arbitrary means but are independent. | Source Title: | Annals of Probability | URI: | http://scholarbank.nus.edu.sg/handle/10635/104154 | ISSN: | 00911798 |
Appears in Collections: | Staff Publications |
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