Benassi, Albert; Cohen, Serge; Istas, Jacques; Jaffard, Stéphane Identification of filtered white noises. (English) Zbl 0932.60037 Stochastic Processes Appl. 75, No. 1, 31-49 (1998). Summary: A class of Gaussian processes, having locally the same fractal properties as fractional Brownian motion, is studied. Our aim is to give estimators of the relevant parameters of these processes from one sample path. A time dependency of the integrand of the classical Wiener integral, associated with the fractional Brownian motion, is introduced. We show how to identify the asymptotic expansion for high frequencies of these integrands on one sample path. Then, the identification of the first terms of this expansion is used to solve some filtering problems. Futhermore, rates of convergence of the estimators are then given. Cited in 1 ReviewCited in 29 Documents MSC: 60G15 Gaussian processes 60G17 Sample path properties 62G05 Nonparametric estimation Keywords:Gaussian processes; identification × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Adler, R.; Pyke, R., Uniform quadratic variation for gaussian processes, Stochastic Process Appl., 48, 191-209 (1993) · Zbl 0783.60040 [2] Benassi, A.; Cohen, S.; Jaffard, S., Identification de processus gaussiens elliptiques, C.R. Acad. Sci. Paris Sér. I, 319, 877-880 (1994) · Zbl 0820.60029 [3] Czörgo, M.; Révész, P., Strong Approximation in Probability and Statistics (1981), Academic Press: Academic Press New York · Zbl 0539.60029 [4] Dacunha-Castelle, D., Duflo, M., 1983. Probabilités et Statistiques, Vol. 2. Masson, Paris.; Dacunha-Castelle, D., Duflo, M., 1983. Probabilités et Statistiques, Vol. 2. Masson, Paris. · Zbl 0535.62004 [5] Goryainov, A., 1998. On Lévy-Baxter theorems for stochastic elliptic equations. Theory Prob. Appl. 33, 164-167.; Goryainov, A., 1998. On Lévy-Baxter theorems for stochastic elliptic equations. Theory Prob. Appl. 33, 164-167. · Zbl 0673.60062 [6] Grenander, U., Abstract Inference (1981), Wiley: Wiley New York · Zbl 0505.62069 [7] Guyon, X., Variations de champs gaussiens stationnaires: applicationà l′identification, Probab. Theory Related Fields, 75, 179-194 (1987) · Zbl 0596.60051 [8] Guyon, X.; Leon, J., Convergence en loi des h−variations d′un processus gaussien stationnaire, Ann. Inst. Poincaré, 25, 265-282 (1989) · Zbl 0691.60017 [9] Istas, J., Estimating the singularity function of a gaussian process with applications, Scand. J. Statist., 23, 5, 581-596 (1996) · Zbl 0898.62106 [10] Istas, J.; Lang, G., Variations quadratiques et estimation de l′exposant de Holder local d′un processus gaussien, C.R. Acad. Sci. Paris Sér. I, 319, 201-206 (1994) · Zbl 0803.60038 [11] Leon, J., Ortega, J., 1989. Weak convergence of different types of variation for biparametric gaussian processes. Colloq. Math. Soc. Janos Bolayi. 57, 349-364. Limit theorems in Probability and Statistics, Pecs Hungary.; Leon, J., Ortega, J., 1989. Weak convergence of different types of variation for biparametric gaussian processes. Colloq. Math. Soc. Janos Bolayi. 57, 349-364. Limit theorems in Probability and Statistics, Pecs Hungary. · Zbl 0727.60040 [12] Mallat S., P.G., Z., Z., 1996. Adaptive covariance estimation of locally stationary process, preprint.; Mallat S., P.G., Z., Z., 1996. Adaptive covariance estimation of locally stationary process, preprint. [13] Mandelbrot, B.; Van Ness, J., Fractional Brownian motions, fractional noises and applications, SIAM Review, 10, 422-437 (1968) · Zbl 0179.47801 [14] Pentland, A., Fractal-based description of natural scenes, IEEE Trans. PAMI., 6, 661-674 (1984) [15] Priestley, M., Evolutionary spectra and non stationary processes, J. Roy. Statist. Soc. B, 27, 204-237 (1965) · Zbl 0144.41001 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.