Choi, K. P.; Klass, Michael J. Some best possible prophet inequalities for convex functions of sums of independent variates and unordered martingale difference sequences. (English) Zbl 0880.60017 Ann. Probab. 25, No. 2, 803-811 (1997). Summary: Let \(\Phi(\cdot)\) be a nondecreasing convex function on \([0,\infty)\). We show that for any integer \(n\geq 1\) and real \(a\), \[ E\Phi((M_n- a)^+)\leq 2E\Phi((S_n- a)^+)- \Phi(0)\quad \text{ and }\quad E(M_n\vee\text{med }S_n)\leq E|S_n-\text{med }S_n|, \] where \(X_1,X_2,\dots\) are any independent mean zero random variables with partial sums \(S_0=0\), \(S_k= X_1+\cdots+ X_k\) 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 \(E(X_k\mid\{X_i: i\neq k\})=0\) a.s. for each \(k\geq 1\). Modified versions of these inequalities hold if the variates have arbitrary means but are independent. Cited in 2 Documents MSC: 60E15 Inequalities; stochastic orderings 60G50 Sums of independent random variables; random walks 60G40 Stopping times; optimal stopping problems; gambling theory 60G42 Martingales with discrete parameter Keywords:maximum of partial sums; sums of independent random variables; prophet inequalities; median; unordered martingale difference sequence; convex function PDF BibTeX XML Cite \textit{K. P. Choi} and \textit{M. J. Klass}, Ann. Probab. 25, No. 2, 803--811 (1997; Zbl 0880.60017) Full Text: DOI OpenURL References: [1] Hill, T. P. and Kertz, R. P. (1992). A survey of prophet inequalities in optimal stopping theory. Contemp. Math. 125 191-207. · Zbl 0794.60040 [2] Klass, M. J. (1989). Maximizing E max1 k n S+k/ES+n: a prophet inequality for sums of i.i.d. mean zero variates. Ann. Probab. 17 1243-1247. · Zbl 0684.60032 [3] Klass, M. J. (1993). Ratio prophet inequalities for convex functions of partial sums. Statist. Probab. Lett. 17 205-209. · Zbl 0777.60038 [4] Klass, M. J. and Teicher, H. (1977). Iterated logarithm laws for asymmetric random variables barely with or without finite mean. Ann. Probab. 5 861-874. · Zbl 0372.60042 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.