On normal domination of (super)martingales

Let (S0, S1, . . . ) be a supermartingale relative to a nondecreasing sequence of σ-algebras (H60,H61, . . . ), with S0 6 0 almost surely (a.s.) and differences Xi := Si − Si−1. Suppose that for every i = 1, 2, . . . there exist H6(i−1)-measurable r.v.’s Ci−1 and Di−1 and a positive real number si such that Ci−1 6 Xi 6 Di−1 and Di−1−Ci−1 6 2si a.s. Then for all natural n and all functions f satisfying certain convexity conditions Ef(Sn) 6 Ef(sZ), where s := √ s1 + · · ·+ sn and Z ∼ N(0, 1). In particular, this implies P(Sn > x) 6 c5,0P(sZ > x) ∀x ∈ R, where c5,0 = 5!(e/5) = 5.699 . . . . Results for max06k6n Sk in place of Sn and for concentration of measure also follow