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](Snn)[less-than-or-equals, slant]cn-p/2, which is...