An Inequality for Tail Probabilities of Martingales with Bounded Differences

Let M_n =X₁+...+X_n be a martingale with bounded differences X_m=M_m-M_m-1 such that {|X_m| _m}=1 with some nonnegative _m. Write ²= ₁²+ ... +_n². We prove the inequalities {M_nx}c(1-(x/)), {M_n x} 1- c(1- (-x/)) with a constant . The result yields sharp inequalities in some models related to the measure concentration phenomena.