A martingale inequality and large deviations

Let (Xi) be a martingale difference sequence and let Sn=[summation operator]i=1nXi. Suppose (Xi) is bounded in Lp. In the case p[greater-or-equal, slanted]2, Lesigne and Volný (Stochastic Process. Appl. 96 (2001) 143) obtained the estimation [mu](Sn>n)[less-than-or-equals, slant]cn-p/2, which is optimal in a certain sense. In this article, we show that [mu](Sn>n)[less-than-or-equals, slant]cn1-p when p[set membership, variant](1,2]. This is optimal for an i.i.d. sequence, as shown in Lesigne and Volný (Stochastic Process. Appl. 96 (2001) 143). For this purpose, we establish some inequalities for (Xi), which may be of interest on their own right.