On Hoeffding's inequalities
| dc.creator | Bentkus, Vidmantas | |
| dc.date | 2004-10-06 | |
| dc.date.accessioned | 2026-07-07T05:12:58Z | |
| dc.date.available | 2026-07-07T05:12:58Z | |
| dc.description | 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_n\geq x}\leq cP{S_n\geq x}, where c is an absolute constant and S_n=ε_1+... +ε_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 x\in R where the survival function x\mapsto P{S_n\geq 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_n\geq 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. | |
| dc.description | Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000360 | |
| dc.identifier | https://arxiv.org/abs/math/0410159 | |
| dc.identifier | http://arxiv.org/abs/math/0410159 | |
| dc.identifier | Annals of Probability 2004, Vol. 32, No. 2, 1650-1673 | |
| dc.identifier | doi:10.1214/009117904000000360 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72778 | |
| dc.subject | Probability | |
| dc.subject | 60E15 (Primary) | |
| dc.title | On Hoeffding's inequalities | |
| dc.type | text |