Probability and moment inequalities for sums of weakly dependent random variables, with applications

Stochastic Processes and their Applications

Volumn 117, Issue 7, 2007, Pages 878-903

  Doukhan, Paul ^a,b   Neumann, Michael H ^c  

^a CREST   (France)

^b University Paris I Sorbonne Panthéon   (France)

^c FRIEDRICH SCHILLER UNIVERSITY JENA   (Germany)

Author keywords

Bernstein inequality; Cumulants; Rosenthal inequality; Weak dependence

Indexed keywords

APPROXIMATION THEORY; ESTIMATION; MATHEMATICAL OPERATORS; PROBLEM SOLVING; RANDOM PROCESSES; STATISTICAL METHODS;

BERNSTEIN INEQUALITY; CUMULANT ESTIMATES; CUMULANTS; INEQUALITIES; ROSENTHAL INEQUALITY; WEAK DEPENDENCE CONDITIONS;

RANDOM VARIABLES;

EID: 34248594720     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2006.10.011     Document Type: Article

References (29)

1. Ango Nze P., Bühlmann P., and Doukhan P. Nonparametric regression estimation under weak dependence beyond mixing and association. Ann. Statist. 30 (2002) 397-430
2. Bennett G. Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 (1962) 33-45
3. Bentkus R., and Rudzkis R. On exponential estimates of the distribution of random variables. Lithuanian Math. J. 20 (1980) 15-30 (in Russian)
4. Borovkova S.A., Burton R.M., and Dehling H.G. Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation. Trans. Amer. Math. Soc. 353 (2001) 4261-4318
5. Butucea C., and Neumann M.H. Exact asymptotics for estimating the marginal density of discretely observed diffusion processes. Bernoulli 11 (2005) 411-444
6. Coulon-Prieur C., and Doukhan P. A triangular CLT for weakly dependent sequences. Statist. Probab. Lett. 47 (2000) 61-68
7. Dedecker J., and Doukhan P. A new covariance inequality and applications. Stochastic Process. Appl. 106 (2003) 63-80
8. Dedecker J., and Prieur C. Coupling for τ-dependent sequences and applications. J. Theoret. Probab. 17 (2004) 861-885
9. Dedecker J., and Prieur C. New dependence coefficients. Examples and applications to statistics. Probab. Theory Related Fields 132 (2005) 203-236
10. Doukhan P. Mixing: Properties and Examples. Lecture Notes in Statistics vol. 85 (1994), Springer-Verlag
11. Doukhan P., and Louhichi S. A new weak dependence condition and application to moment inequalities. Stochastic Process. Appl. 84 (1999) 313-342
12. Doukhan P., and Louhichi S. Functional estimation for weakly dependent stationary time series. Scand. J. Statist. 28 (2001) 325-342
13. Doukhan P., Teyssière G., and Winant P. Vector valued LARCH infinity processes. In: Bertail P., Doukhan P., and Soulier P. (Eds). Dependence in Probability and Statistics (2005), Springer, New York
14. P. Doukhan, O. Wintenberger, A central limit theorem under non causal weak dependence conditions and sharp moment assumptions, 2005. preprint
15. Giné E., and Guillou A. Rates of strong uniform consistency for multivariate kernel density estimators. Ann. Inst. H. Poincaré 38 (2002) 907-921
16. Giraitis L., Leipus L., and Surgailis D. Recent advances in ARCH modelling. In: Teyssière G., and Kirman A. (Eds). Long Memory in Economics (2003), Springer Verlag
17. Hall P., and Heyde C.C. Martingale Limit Theory and its Applications (1980), Academic Press, New York
18. Johnson B.W., Schechtman G., and Zinn J. Best constants in moment inequalities for linear combinations of independent and exchangeable random variables. Ann. Probab. 13 (1985) 234-253
19. Kallabis R.S., and Neumann M.H. An exponential inequality under weak dependence. Bernoulli 12 (2006) 333-350
20. Korostelev A.P., and Nussbaum M. The asymptotic minimax constant for sup-norm loss in nonparametric density estimation. Bernoulli 5 (1999) 1099-1118
21. Móricz F.A., Serfling J.A., and Stout W.F. Moment and probability bounds with quasi-superadditive structure for the maximum partial sum. Ann. Probab. 10 (1982) 1032-1040
22. M.H. Neumann, E. Paparoditis, Goodness-of-fit tests for Markovian time series models, Technical Report No. 16/2005, Department of Mathematics and Statistics, University of Cyprus, 2005
23. N. Ragache, O. Wintenberger, Convergence rates for density estimates of weakly dependent time series, SAMOS, Paris 1, 2005. preprint
24. Rio E. Théorie asymptotique pour des processus aléatoires faiblement dépendants. SMAI, Mathématiques et Applications vol. 31 (2000), Springer
25. p (p > 2) spanned by sequences of independent random variables. Israel J. Math. 8 (1970) 273-303
26. Rota G.C., and Chen J. On the combinatorics of cumulants. J. Combin. Theory Ser. A 91 (2000) 283-304
27. Saulis L., and Statulevicius V.A. Limit Theorems for Large Deviations, Mathematics and its Applications (1991), Kluwer
28. Statulevicius V.A. Limit theorems for random functions. Lithuanian Math. J. 10 (1970) 583-592 (in Russian)
29. Talagrand M. The Generic Chaining (2005), Springer-Verlag