Abstract
Let ( X i ) be a martingale difference sequence and let S n =∑ i=1 n X i . Suppose ( X i ) is bounded in L p . In the case p⩾2, Lesigne and Volný (Stochastic Process. Appl. 96 (2001) 143) obtained the estimation μ( S n > n)⩽ cn − p/2 , which is optimal in a certain sense. In this article, we show that μ( S n > n)⩽ cn 1− p when p∈(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 ( X i ), which may be of interest on their own right.
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