Binomial upper bounds on generalized moments and tail probabilities of (super)martingales with differences bounded from above

Let (S0, S1, . . . ) be a supermartingale relative to a nondecreasing sequence of σ-algebras H≤0, H≤1, . . . , with S0 ≤ 0 almost surely (a.s.) and differences Xi := Si − Si−1. Suppose that Xi ≤ d and Var(Xi|H≤i−1) ≤ σ 2 i a.s. for every i = 1, 2, . . . , where d > 0 and σi > 0 are non-random constants. Let Tn := Z1 + · · · + Zn, where Z1, . . . , Zn are i.i.d. r.v.’s each taking on only two values, one of which is d, and satisfying the conditions EZi = 0 and VarZi = σ := 1 n (σ 1 + · · · + σ n). Then, based on a comparison inequality between generalized moments of Sn and Tn for a rich class of generalized moment functions, the tail comparison inequality P(Sn ≥ y) ≤ cP (Tn ≥ y + h 2 ) ∀y ∈ R is obtained, where c := e/2 = 3.694 . . . , h := d + σ/d, and the function y 7→ P(Tn ≥ y) is the least log-concave majorant of the linear interpolation of the tail function y 7→ P(Tn ≥ y) over the lattice of all points of the form nd+kh (k ∈ Z). An explicit formula for P(Tn ≥ y+ h 2 ) is given. Another, similar bound is given under somewhat different conditions. It is shown that these bounds improve significantly upon known bounds.