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A martingale inequality and large deviations

✍ Scribed by Yulin Li

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
103 KB
Volume
62
Category
Article
ISSN
0167-7152

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✦ Synopsis

Let (X i ) be a martingale di erence sequence and let S n = n i=1 X i . Suppose (X i ) is bounded in L p . In the case p ¿ 2, Lesigne and Volnà y (Stochastic Process. Appl. 96 (2001) 143) obtained the estimation (S n ¿ n) 6 cn -p=2 , which is optimal in a certain sense. In this article, we show that (S n ¿ n) 6 cn 1-p when p ∈ (1; 2]. This is optimal for an i.i.d. sequence, as shown in Lesigne and Volnà y (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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