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Probability and moment inequalities for sums of weakly dependent random variables, with applications. (English) Zbl 1117.60018

Summary: P. Doukhan and S. Louhichi [Stochastic Processes Appl. 84, 313–342 (1999; Zbl 0996.60020)] introduced a new concept of weak dependence which is more general than mixing. Such conditions are particularly well suited for deriving estimates for the cumulants of sums of random variables. We employ such cumulant estimates to derive inequalities of Bernstein and Rosenthal type which both improve on previous results. Furthermore, we consider several classes of processes and show that they fulfill appropriate weak dependence conditions. We also sketch applications of our inequalities in probability and statistics.

MSC:

60E15 Inequalities; stochastic orderings
62E99 Statistical distribution theory

Citations:

Zbl 0996.60020
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References:

[1] Ango Nze, P.; Bühlmann, P.; Doukhan, P., Nonparametric regression estimation under weak dependence beyond mixing and association, Ann. Statist., 30, 397-430 (2002) · Zbl 1012.62037
[2] Bennett, G., Probability inequalities for the sum of independent random variables, J. Amer. Statist. Assoc., 57, 33-45 (1962) · Zbl 0104.11905
[3] Bentkus, R.; Rudzkis, R., On exponential estimates of the distribution of random variables, Lithuanian Math. J., 20, 15-30 (1980), (in Russian) · Zbl 0428.60027
[4] Borovkova, S. A.; Burton, R. M.; Dehling, H. G., Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation, Trans. Amer. Math. Soc., 353, 4261-4318 (2001) · Zbl 0980.60031
[5] Butucea, C.; Neumann, M. H., Exact asymptotics for estimating the marginal density of discretely observed diffusion processes, Bernoulli, 11, 411-444 (2005) · Zbl 1069.62062
[6] Coulon-Prieur, C.; Doukhan, P., A triangular CLT for weakly dependent sequences, Statist. Probab. Lett., 47, 61-68 (2000) · Zbl 0956.60006
[7] Dedecker, J.; Doukhan, P., A new covariance inequality and applications, Stochastic Process. Appl., 106, 63-80 (2003) · Zbl 1075.60513
[8] Dedecker, J.; Prieur, C., Coupling for \(\tau \)-dependent sequences and applications, J. Theoret. Probab., 17, 861-885 (2004) · Zbl 1067.60008
[9] Dedecker, J.; Prieur, C., New dependence coefficients. Examples and applications to statistics, Probab. Theory Related Fields, 132, 203-236 (2005) · Zbl 1061.62058
[10] Doukhan, P., (Mixing: Properties and Examples. Mixing: Properties and Examples, Lecture Notes in Statistics, vol. 85 (1994), Springer-Verlag) · Zbl 0801.60027
[11] Doukhan, P.; Louhichi, S., A new weak dependence condition and application to moment inequalities, Stochastic Process. Appl., 84, 313-342 (1999) · Zbl 0996.60020
[12] Doukhan, P.; Louhichi, S., Functional estimation for weakly dependent stationary time series, Scand. J. Statist., 28, 325-342 (2001)
[13] Doukhan, P.; Teyssière, G.; Winant, P., Vector valued LARCH infinity processes, (Bertail, P.; Doukhan, P.; Soulier, P., Dependence in Probability and Statistics (2005), Springer: Springer New York)
[14] P. Doukhan, O. Wintenberger, A central limit theorem under non causal weak dependence conditions and sharp moment assumptions, 2005. preprint; P. Doukhan, O. Wintenberger, A central limit theorem under non causal weak dependence conditions and sharp moment assumptions, 2005. preprint
[15] Giné, E.; Guillou, A., Rates of strong uniform consistency for multivariate kernel density estimators, Ann. Inst. H. Poincaré, 38, 907-921 (2002) · Zbl 1011.62034
[16] Giraitis, L.; Leipus, L.; Surgailis, D., Recent advances in ARCH modelling, (Teyssière, G.; Kirman, A., Long Memory in Economics (2003), Springer Verlag) · Zbl 1180.62121
[17] Hall, P.; Heyde, C. C., Martingale Limit Theory and its Applications (1980), Academic Press: Academic Press New York · Zbl 0462.60045
[18] Johnson, B. W.; Schechtman, G.; Zinn, J., Best constants in moment inequalities for linear combinations of independent and exchangeable random variables, Ann. Probab., 13, 234-253 (1985) · Zbl 0564.60020
[19] Kallabis, R. S.; Neumann, M. H., An exponential inequality under weak dependence, Bernoulli, 12, 333-350 (2006) · Zbl 1126.62039
[20] Korostelev, A. P.; Nussbaum, M., The asymptotic minimax constant for sup-norm loss in nonparametric density estimation, Bernoulli, 5, 1099-1118 (1999) · Zbl 0955.62037
[21] Móricz, F. A.; Serfling, J. A.; Stout, W. F., Moment and probability bounds with quasi-superadditive structure for the maximum partial sum, Ann. Probab., 10, 1032-1040 (1982) · Zbl 0499.60052
[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; 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 · Zbl 1155.62058
[23] N. Ragache, O. Wintenberger, Convergence rates for density estimates of weakly dependent time series, SAMOS, Paris 1, 2005. preprint; N. Ragache, O. Wintenberger, Convergence rates for density estimates of weakly dependent time series, SAMOS, Paris 1, 2005. preprint · Zbl 1113.62055
[24] Rio, E., Théorie asymptotique pour des processus aléatoires faiblement dépendants, (SMAI, Mathématiques et Applications, vol. 31 (2000), Springer) · Zbl 0944.60008
[25] Rosenthal, H. P., On the subspaces of \(L^p(p > 2)\) spanned by sequences of independent random variables, Israel J. Math., 8, 273-303 (1970) · Zbl 0213.19303
[26] Rota, G. C.; Chen, J., On the combinatorics of cumulants, J. Combin. Theory Ser. A, 91, 283-304 (2000) · Zbl 0978.05006
[27] Saulis, L.; Statulevicius, V. A., Limit Theorems for Large Deviations, Mathematics and its Applications (1991), Kluwer · Zbl 0810.60024
[28] Statulevicius, V. A., Limit theorems for random functions, Lithuanian Math. J., 10, 583-592 (1970), (in Russian) · Zbl 0266.60014
[29] Talagrand, M., The Generic Chaining (2005), Springer-Verlag · Zbl 1075.60001
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