On normal domination of (super)martingales

Document Type

Article

Publication Date

1-1-2006

Abstract

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 for every i = 1, 2, . . . there exist H≼(i-1)-measurable r.v.’s Ci-1 and Di-1 and a positive real number si such that Ci-1 ≼ Xi ≼ Di-1 and Di-1 - Ci-1 ≼ 2si a.s. Then for all natural n and all functions f satisfying certain convexity conditions Ef(Sn ≼ Ef(sZ), where s := √ s21 + . . . + s2n n and Z ∼ N(0, 1). In particular, this implies P(Sn ≽ x) ≼ c5, 0 P(sZ ≽ x) ∀x ∈ ℝ, where c5, 0 = 5!(e/5)5 = 5.699 . . . . Results for max0≼k≼n Sk in place of Sn and for concentration of measure also follow. © 2006 Applied Probability Trust.

Publication Title

Electronic Journal of Probability

Link to Full Text

Share

COinS