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On Hoeffding's inequalities | | Vidmantas Bentkus
; | | Date: |
6 Oct 2004 | | Journal: | Annals of Probability 2004, Vol. 32, No. 2, 1650-1673 DOI: 10.1214/009117904000000360 | | Subject: | Probability MSC-class: 60E15 (Primary) | math.PR | | Abstract: | In a celebrated work by Hoeffding [J. Amer. Statist. Assoc. 58 (1963) 13-30], several inequalities for tail probabilities of sums M_n=X_1+... +X_n of bounded independent random variables X_j were proved. These inequalities had a considerable impact on the development of probability and statistics, and remained unimproved until 1995 when Talagrand [Inst. Hautes Etudes Sci. Publ. Math. 81 (1995a) 73-205] inserted certain missing factors in the bounds of two theorems. By similar factors, a third theorem was refined by Pinelis [Progress in Probability 43 (1998) 257-314] and refined (and extended) by me. In this article, I introduce a new type of inequality. Namely, I show that P{M_ngeq x}leq cP{S_ngeq x}, where c is an absolute constant and S_n=epsilon_1+... +epsilon_n is a sum of independent identically distributed Bernoulli random variables (a random variable is called Bernoulli if it assumes at most two values). The inequality holds for those xin R where the survival function xmapsto P{S_ngeq x} has a jump down. For the remaining x the inequality still holds provided that the function between the adjacent jump points is interpolated linearly or log-linearly. If it is necessary, to estimate P{S_ngeq x} special bounds can be used for binomial probabilities. The results extend to martingales with bounded differences. It is apparent that Theorem 1.1 of this article is the most important. | | Source: | arXiv, math.PR/0410159 | | Services: | Forum | Review | PDF | Favorites | |
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