## 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.

### MSC:

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