Moderate deviations for functional U -processes

Annales de l'institut Henri Poincare (B) Probability and Statistics

Volumn 37, Issue 2, 2001, Pages 245-273

  Eichelsbacher, Peter ^a  

^a BIELEFELD UNIVERSITY   (Germany)

Author keywords

60E15; 60F05 (primary); 60F10; 60G17; Bernstein type inequality; Decoupling inequality; Maximal inequality for U statistics; Moderate deviations; Partial sums; U processes; VC classes

Indexed keywords

EID: 0042334604     PISSN: 02460203     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0246-0203(00)01063-3     Document Type: Article

References (29)

1. Arcones M.A., Giné E. Limit theorems for U -processes. Ann. Probab. 21:1993;1494-1542.
2. Arcones M.A., Giné E. U -processes indexed by Vapnik-Červonenkis classes of functions with applications to asymptotics and bootstrap of U -statistics with estimated parameters. Stochastic Process. Appl. 52:1994;17-38.
3. Arcones M.A., Giné E. On the law of the iterated logarithm for canonical U -statistics and processes. Stochastic Process. Appl. 58:1995;217-245.
4. Baldi P. Large deviations and stochastic homogenization. Ann. Mat. Pura Appl. 151:1988;161-177.
5. Borovkov A.A., Mogulskii A.A. Probabilities of large deviations in topological spaces I. Siberian Math. J. 19:1978;697-709.
6. Borovkov A.A., Mogulskii A.A. Probabilities of large deviations in topological spaces II. Siberian Math. J. 21:1980;653-664.
7. Borovskikh Yu.V. U -Statistics in Banach Spaces. 1996;VSP, Utrecht.
8. Dembo A., Zajic T. Uniform large and moderate deviations for functional empirical processes. Stochastic Process. Appl. 67:1997;195-211.
9. Dembo A., Zeitouni O. Large Deviations Techniques and Applications. 1998;Springer, New York.
10. Dudley R.M. A course on empirical processes. Dold A., Eckmann B. École d'Eté de Probabilités de Saint-Flour XII. Lecture Notes in Math. 1097:1984;1-142 Springer, New York.
11. Eichelsbacher P. Large deviations for products of empirical probability measures in the τ -topology. J. Theoret. Probab. 10:(4):1997;903-920.
12. Eichelsbacher P., Löwe M. Large deviations for partial sums U -processes. Theory Probab. Appl. 43:1998;97-115.
13. Eichelsbacher P., Schmock U. Exponential approximations in completely regular topological spaces and extensions of Sanov's theorem. Stochastic Process. Appl. 77:(2):1998;233-251.
14. Eichelsbacher P., Schmock U. Large and rank-dependent moderate deviations for U -empirical measures in strong topologies, preprint. 1998.
15. Giné E., de la Peña V. Decoupling: From Dependence to Independence. 1999;Springer-Verlag.
16. Giné E., Zhang C.-H. On integrability in the LIL for degenerate U -statistics. J. Theoret. Probab. 9:(2):1996;385-412.
17. Hall P. On the invariance principle for U -statistics. Stochastic Process. Appl. 9:1979;163-174.
18. Hoeffding W. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58:1963;13-30.
19. Ledoux M., Talagrand M. Probability in Banach Spaces. 1991;Springer-Verlag, Berlin.
20. Mandelbaum A., Taqqu M.S. Invariance principle for symmetric statistics. Ann. Statist. 12:1984;483-496.
21. Miller R.G., Sen P.K. Weak convergence of U -statistics and von Mises differentiable. Ann. Math. Statist. 43:1972;31-41.
22. Mogulskii A.A. Large deviations for trajectories of multidimensional random walks. Theory Probab. Appl. 21:1976;300-315.
23. Montgomery-Smith S.J. Comparison of sums of independent identically distributed random vectors. Probab. Math. Statist. 14:(2):1994;281-285.
24. Montgomery-Smith S.J., de la Peña V. Decoupling inequalities for the tail probabilities of multivariate U -statistics. Ann. Probab. 23:(2):1995;806-816.
25. Pollard D. Convergence of Stochastic Processes. 1984;Springer, New York.
26. Talagrand M. Sharper bounds for gaussian and empirical processes. Ann. Probab. 22:(1):1994;28-76.
27. Wu L. Large deviations, moderate deviations and LIL for empirical processes. Ann. Probab. 22:(1):1994;17-27.
28. Yurinsky V.V. Exponential inequalities for sums of random vectors. J. Mult. Anal. 6:(4):1976;473-499.
29. Yurinsky V.V. Sums and Gaussian Vectors. Lecture Notes in Mathematics. 1617:1995;Springer-Verlag, Berlin.