Normal domination of (super)martingales

Abstract

Let (S_0,S_1,...) be a supermartingale relative to a nondecreasing sequence of \sigma-algebras (H_{\le0},H_{\le1},...), with S_0\le0 almost surely (a.s.) and differences X_i:=S_i-S_{i-1}. Suppose that for every i=1,2,... there exist H_{\le(i-1)}-measurable r.v.'s C_{i-1} and D_{i-1} and a positive real number s_i such that C_{i-1}\le X_i\le D_{i-1} and D_{i-1}-C_{i-1}\le 2 s_i a.s. Then for all real t and natural n one has \E f_t(S_n)\le\E f_t(sZ), where f_t(x):=\max(0,x-t)^5, s:=\sqrt{s_1^2+...+s_n^2}, and Z is N(0,1). In particular, this implies P(S_n\ge x)\le c_{5,0}P(Z\ge x/s) for all x in \R, where c_{5,0}=5!(e/5)^5=5.699.... Results for \max_{0\le k\le n}S_k in place of S_n and for concentration of measure also follow.