On Hoeffding's inequalities

Annals of Probability

Volumn 32, Issue 2, 2004, Pages 1650-1673

  Bentkus, Vidmantas ^a,b  

^a VYTAUTAS MAGNUS UNIVERSITY   (Lithuania)

^b VILNIUS UNIVERSITY   (Lithuania)

Author keywords

Bounded differences and random variables; Bounds for tail probabilities; Hoeffding's inequalities; Inequalities; Martingale; Probabilities of large deviations

Indexed keywords

EID: 3042635751     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117904000000360     Document Type: Article

References (34)

1. ALON, N. and MILMAN, V. D. (1984). Concentration of measure phenomena in the discrete case and the Laplace operator of a graph. Publ. Math. Univ. Paris VII 20 55-68.
2. BAHADUR, R. R. (1971). Some Limit Theorems in Statistics. SIAM, Philadelphia.
3. BENTKUS, V. (1994). On the asymptotical behavior of the constant in the Berry-Esseen inequality. J. Theoret. Probab. 7 211-224.
4. BENTKUS, V. (2001). An inequality for large deviation probabilities of sums of bounded i.i.d. r.v. Lithuanian Math. J. 41 144-153.
5. BENTKUS, V. (2003). An inequality for tail probabilities of martingales with differences bounded from one side. J. Theor. Probab. 16 161-173.
6. BENTKUS, V. (2004). On measure concentration for separately Lipschitz functions in product spaces. Israel J. Math. To appear.
7. BENTKUS, V. and KIRSHA, K. (1989). Estimates for the closeness of a distribution function to the normal law. Lithuanian Math. J. 29 321-332.
8. BENTKUS, V. and VAN ZUIJLEN, M. (2003). Conservative confidence bounds for the mean Lithuanian Math. J. 43 141-160.
9. BRETAGNOLLE, J. (1980). Statistique de Kolmogorov-Smirnov pour un enchantillon nonequireparti. Colloqvium Internationale CNRS 307 39-44.
10. EATON, M. L. (1970). A note on symmetric Bernoulli random variables. Ann. Math. Statist. 41 1223-1226.
11. EATON, M. L. (1974). A probability inequality for linear combinations of bounded random variables. Ann. Statist. 2609-613.
12. GLESER, L. J. (1975). On the distribution of the number of successes in independent trials. Ann. Probab. 3 182-188.
13. GROMOV, M. and MILMAN, V. D. (1983). A topological application of the isoperimetric inequality. Amer. J. Math. 105 843-854.
14. HOEFFDING, W. (1956). On the distribution of the number of successes in independent trials. Ann. Math. Statist. 27 713-721.
15. HOEFFDING, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13-30.
16. IBRAGIMOV, I. A. (1956). On the composition of unimodal distributions. Theory Probab. Appl. 1 255-260.
17. KARLIN, S. and STUDDEN, W. J. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics. Wiley, New York.
18. KEILSON, J. and GERBER, H. (1971). Some results for discrete unimodality. J. Amer. Statist. Assoc. 66 386-389.
19. LEDOUX, M. (1999). Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités XXXIII. Lecture Notes in Math. 1709 120-216. Springer, Berlin.
20. MARSHALL, A. W. and OLKIN, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
21. MCDIARMID, C. (1989). On the method of bounded differences. In Surveys in Combinatorics. London Math. Soc. Lecture Note Ser. 141 148-188. Cambridge Univ. Press.
22. MILMAN, V. D. (1985). Random subspaces of proportional dimension of finite-dimensional normed spaces: Approach through the isoperimetric inequality. Lecture Notes in Math. 1166 106-115. Springer, Berlin.
23. MILMAN, V. D. (1988). The heritage of P. Lévy in geometrical functional analysis. Astérisque 157-158 273-301.
24. MILMAN, V. D. and SCHECHTMAN, G. (1986). Asymptotic theory of finite-dimensional normed spaces. Lecture Notes in Math. 1200. Springer, Berlin.
25. OSTROWSKI, A. (1952). Sur quelques applications des fonctions convexes et concave au sens de I. Schur. J. Math. Pures Appl. 31 253-292.
26. PAULAUSKAS, V. (2002), Some comments on deviation probabilities for infinitely divisible random vectors. Lithuanian Math. J. 42 394-410.
27. PETROV, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
28. 2 test under a symmetry assumption. Ann. Statist. 22 357-368.
29. PINELIS, I. (1998). Optimal tail comparison based on comparison of moments. In High Dimensional Probability. Progress in Probability 43 297-314. Birkhäuser, Basel.
30. PINELIS, I. (1999). Fractional sums and integrals of r-concave tails and applications to comparison probability inequalities. In Advances in Stochastic Inequalities. Contemporary Mathematics 234 149-168. Amer. Math. Soc. Providence, RI.
31. SCHUR, I. (1923). Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinanten-theorie. Sitzungsber. Berlin Math. Ges. 22 9-20.
32. SHORACK, G. R. and WELLNER, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
33. TALAGRAND, M. (1995a). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81 73-205.
34. TALAGRAND, M. (1995b). The missing factor in Hoeffding's inequalities. Ann. Inst. H. Poincaré Probab. Statist. 31 689-702.