On normal domination of (super)martingales. (English) Zbl 1130.60019

Summary: Let \((S_0, S_1,\dots )\) be a supermartingale relative to a nondecreasing sequence of \(\sigma\)-algebras \((H_{\leq0},H_{\leq 1},\dots)\), with \(S_0\leq 0\) almost surely (a.s.) and differences \(X_i := S_i-S_{i-1}\). Suppose that for every \(i = 1, 2,\dots\) there exist \(H_{\leq(i-1)}\)-measurable random variables \(C_{i-1}\) and \(D_{i-1}\) and a positive real number \(s_i\) such that \(C_{i-1}\leq X_i\leq D_{i-1}\) and \(D_{i-1}-C_{i-1}\leq 2s_i\) a.s. Then for all natural \(n\) and all functions \(f\) satisfying certain convexity conditions
\[ {\mathbf E} f(S_n)\leq {\mathbf E}f(sZ), \]
where \(s := \sqrt{s^2_1+\cdots + s_n^2}\) and \(Z\sim N(0, 1)\). In particular, this implies
\[ P(S_n\geq x)\leq c_{5,0}{\mathbf P}(sZ\geq x)\quad \forall x\in\mathbb R, \]
where \(c_{5,0}= 5!(e/5)^5 = 5.699\dots\,\). Results for \(\max_{0\leq k\leq n} S_k\) in place of \(S_n\) and for concentration of measure also follow.


60E15 Inequalities; stochastic orderings
60E10 Characteristic functions; other transforms
60G42 Martingales with discrete parameter
60F10 Large deviations
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