An Inequality for Tail Probabilities of Martingales with Bounded Differences

Abstract

Let M n =X1+...+Xn be a martingale with bounded differences Xm=Mm-Mm-1 such that ℙ{|Xm|≤ σ m}=1 with some nonnegative σm. Write σ2= σ 1 2 + ... +σ n 2 . We prove the inequalities ℙ{M n≥x}≤c(1-Φ(x/σ)), ℙ{M n ⩾ x}⩽ 1- c(1-Φ (-x/σ)) with a constant $$c \leqslant 1(1 - \Phi (\sqrt 3 )) \leqslant 25$$ . The result yields sharp inequalities in some models related to the measure concentration phenomena.