×

Some laws of the iterated logarithm in Hilbertian autoregressive models. (English) Zbl 1142.60323

Summary: We consider the law of the iterated logarithm for the empirical covariance of Hilbertian autoregressive processes. As an application, we obtain laws of the iterated logarithm for the eigenvalues and associated projectors of the empirical covariance.

MSC:

60F15 Strong limit theorems
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
62M99 Inference from stochastic processes

Software:

fda (R)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Besse, H.; Cardot, H., Approximation spline de la prévision d’un processus fonctionnel autorégressif d’ordre 1, Canad. J. Statist, 24, 467-487 (1996) · Zbl 0879.62092
[2] Besse, H.; Cardot, H.; Stephenson, D., Autoregressive forecasting of some climatic variations, Scand. J. Statist, 27, 673-687 (2000) · Zbl 0962.62089
[3] D. Bosq Modelization, nonparametric estimation and prediction for continuous time processes, in: Nonparametric Functional Estimation and Related Topics, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 335, Kluwer Acad. Publ., Dordrecht, 1991, pp. 509-529.; D. Bosq Modelization, nonparametric estimation and prediction for continuous time processes, in: Nonparametric Functional Estimation and Related Topics, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 335, Kluwer Acad. Publ., Dordrecht, 1991, pp. 509-529.
[4] Bosq, D., Linear Processes in Function Spaces, Lecture Notes in Statistics, Vol. 149 (2000), Springer: Springer Berlin · Zbl 0971.62023
[5] Bosq, D., Estimation of mean and covariance operator of autoregressive processes in Banach spaces, Statist. Inference Stochastic Process, 5, 3, 287-306 (2002) · Zbl 1028.62070
[6] A. Cavallini, G.C. Montanari, M. Loggini, O. Lessi, M. Cacciari, Nonparametric prediction of harmonic levels in electrical networks, Proceedings of I.E.E.E. I.C.H.P.S VI, Bologna, 1994, pp. 165-171.; A. Cavallini, G.C. Montanari, M. Loggini, O. Lessi, M. Cacciari, Nonparametric prediction of harmonic levels in electrical networks, Proceedings of I.E.E.E. I.C.H.P.S VI, Bologna, 1994, pp. 165-171.
[7] Csado, C.; Taqqu, M. S., A survey of functional laws of the iterated logarithm for self-similar processes, Comm. Statist. Stochastic Models, 1, 77-115 (1985)
[8] Chen, X., The law of the iterated logarithm for m-dependent Banach space valued random variables, J. Theorit. Probab, 10, 3, 695-732 (1997) · Zbl 0883.60008
[9] Dauxois, J.; Pousse, A.; Romain, Y., Asymptotic theory for the principal component analysis of a vector random functionsome applications to statistical inference, J. Multivariate Anal, 12, 136-154 (1982) · Zbl 0539.62064
[10] Dehling, H.; Denker, M.; Philipp, W., A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics, Probab. Theory Related Fields, 72, 111-131 (1986) · Zbl 0572.60012
[11] Deuschel, J. D.; Stroock, D. W., Large deviations, Pure and Applied Mathematics, Vol. 137 (1989), Academic Press: Academic Press Boston, MA · Zbl 0682.60018
[12] Diestel, J.; Uhl, J. J., Vector Measures, Mathematical Survey of the A.M.S, Vol. 15 (1977), Amer. Math. Soc: Amer. Math. Soc Providence, RI · Zbl 0369.46039
[13] N. Dunford, J.T. Schwartz, Linear Operators, Vol. II, Wiley Classics Library, Wiley, New York, 1958.; N. Dunford, J.T. Schwartz, Linear Operators, Vol. II, Wiley Classics Library, Wiley, New York, 1958. · Zbl 0084.10402
[14] Ledoux, M.; Talagrand, M., Probability in Banach Spaces (1991), Springer: Springer Berlin · Zbl 0662.60008
[15] A. Mas, L. Menneteau, Perturbation approach applied to the asymptotic study of random operators, in: High Dimensional Probability III, Progr. Probab., Birkhauser, Boston, 2003.; A. Mas, L. Menneteau, Perturbation approach applied to the asymptotic study of random operators, in: High Dimensional Probability III, Progr. Probab., Birkhauser, Boston, 2003. · Zbl 1053.60002
[16] Mas, A.; Menneteau, L., Large and moderate deviations for infinite dimensional autoregressive processes, J. Multivar. Anal, 87, 241-260 (2003) · Zbl 1043.60021
[17] Ramsay, J. O.; Silverman, B. W., Functional Data Analysis, Springer Series in Statistics (1997), Springer: Springer Berlin · Zbl 0882.62002
[18] Ruymgaart, F. H.; Yang, S., Some applications of Watson’s perturbation approach to random matrices, J. Multivariate Anal, 60, 48-60 (1997), doi:10.1006/jmva.l996.1640 · Zbl 0927.62018
[19] Yurinskii, V. V., Exponential inequalities for sum of random vectors, J. Multivariate Anal, 6, 473-499 (1976) · Zbl 0346.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.