Some best possible prophet inequalities for convex functions of sums of independent variates and unordered martingale difference sequences. (English) Zbl 0880.60017

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.


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
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