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Wavelet analysis and covariance structure of some classes of non-stationary processes. (English) Zbl 0958.60034
Summary: Processes with stationary \(n\)-increments are known to be characterized by the stationarity of their continuous wavelet coefficients. We extend this result to the case of processes with stationary fractional increments and locally stationary processes. Then we give two applications of these properties. First, we derive the explicit covariance structure of processes with stationary \(n\)-increments. Second, for fractional Brownian motion, the stationarity of the fractional increments of order greater than the Hurst exponent is recovered.

MSC:
60G12 General second-order stochastic processes
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
46N30 Applications of functional analysis in probability theory and statistics
28A80 Fractals
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References:
[1] Abry, P., Flandrin, P., Taqqu, M.S., and Veitch, D. (1999). Wavelets for the analysis, estimation and synthesis of scaling data, InSelf-Similar Network Traffic and Performance Evaluation, to appear.
[2] Averkamp, R. and Houdré, C. A. note on the discrete wavelet transform of second-order processes,IEEE Trans. Inform. Theory, to appear. · Zbl 0996.60051
[3] Averkamp, R. and Houdré, C. (1998). Some distributional properties of the continuous wavelet transform of random processes,IEEE Trans. Inform. Theory,44(3), 1111–1124. · Zbl 0906.60033 · doi:10.1109/18.669179
[4] Brockwell, P.J. and Davis, R.A. (1987).Time Series: Theory and Methods, Springer-Verlag, New York. · Zbl 0604.62083
[5] Cambanis, S. and Houdré, C. (1995). On the continuous wavelet transform of second-order random processes,IEEE Trans. Inform. Theory,41(3), 628–642. · Zbl 0820.60022 · doi:10.1109/18.382010
[6] Cheng, B. and Tong, H. (1996).A Theory of Wavelet Representation and Decomposition for a General Stochastic Process, Number 115 in Lectures Notes in Statistics. Springer-Verlag, New York, 115–129.
[7] Dijkerman, R. W. and Mazumdar, R.R. (1994). On the correlation structure of the wavelet coefficients of fractional brownian motion,IEEE Trans. Inform. Theory,40(5), 1609–1616. · Zbl 0810.60079 · doi:10.1109/18.333875
[8] Dijkerman, R.W. and Mazumdar, R.R. (1994). Wavelet representations of stochastic processes and multiresolution stochastic models,IEEE Trans. Signal Proc.,42(7), 1640–1652. · doi:10.1109/78.298272
[9] Doob, J.L. (1953).Stochastic Processes, Wiley Publications in Statistics, Wiley, New York. · Zbl 0053.26802
[10] Flandrin, P. (1989). On the spectrum of fractional brownian motion,IEEE Trans. Inform. Theory,35(1), 197–199. · doi:10.1109/18.42195
[11] Flandrin, P. (1992). Wavelet analysis and synthesis of fractional brownian motion,IEEE Trans. Inform. Theory,38(2), 910–917. · Zbl 0743.60078 · doi:10.1109/18.119751
[12] Folland, G.B. (1984).Real Analysis, John Wiley & Sons, New York. · Zbl 0549.28001
[13] Gel’fand, I.M. (1955). Generalized random processes,Dokl. Akad. Nauk. SSSR,100, 853.
[14] Gel’fand, I.M. and Shilov, G.E. (1964).Generalized Functions. Vol. 1. Academic Press [Harcourt Brace Jovanovich Publishers], New York, [1977]. Properties and operations, Translated from the Russian by Eugene Saletan.
[15] Gel’fand, I.M. and Vilenkin, N.Ya. (1964).Generalized Functions. Vol. 4. Academic Press [Harcourt Brace Jovanovich Publishers], New York, [1977]. Applications of harmonic analysis, Translated from the Russian by Amiel Feinstein.
[16] Granger, C.W.J. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing,J. Time Ser. Anal.,1(1), 15–29. · Zbl 0503.62079 · doi:10.1111/j.1467-9892.1980.tb00297.x
[17] Hosking, J.R.M. (1981). Fractional differencing,Biometrika,68(1), 165–176. · Zbl 0464.62088 · doi:10.1093/biomet/68.1.165
[18] Houdré, C. (1990). Harmonizability, v-boundedness, (2,p)-boundedness of stochastic processes,Probab. Th. Rel. Fields,87, 167–188. · Zbl 0688.60028 · doi:10.1007/BF01198428
[19] Houdré, C. (1993). Wavelets, probability and statistics: some bridges,Wavelets: Mathematics and Applications, Benedetto, J. and Frazier, M., Eds., CRC Press, Boca Raton, FL, 361–394.
[20] Krim, H. and Pesquet, J.C. (1995). Multiresolution analysis of a class of nonstationary processes,IEEE Trans. Inform. Theory,41, 1010–1020. · Zbl 0833.60042 · doi:10.1109/18.391246
[21] Loève, M. (1978).Probability Theory. II. 4th ed., Springer-Verlag, New York, Graduate Texts in Mathematics, Vol. 46.
[22] Mallat, S., Papanicolaou, G., and Zhang, Z. (1998). Adaptive covariance estimation of locally stationary processes,Ann. Statist.,26(1), 1–47. · Zbl 0949.62082 · doi:10.1214/aos/1030563977
[23] Mandelbrot, B.B. and Van Ness, J.W. (1968). Fractional Brownian motions, fractional noises and applications,SIAM Rev.,10, 422–437. · Zbl 0179.47801 · doi:10.1137/1010093
[24] Masry, E. (1993). The wavelet transform to stochastic processes with stationary increments and its application to fractional brownian motion,IEEE Trans. Inform. Theory,39(1), 260–264. · Zbl 0768.60036 · doi:10.1109/18.179371
[25] Masry, E. (1996). Convergence properties of wavelet series expansions of fractional brownian motion.Appl. Comp. Harm. Anal.,3, 239–253. · Zbl 0898.60080 · doi:10.1006/acha.1996.0019
[26] Michálek, J. (1986). Ergodic properties of locally stationary processes,Kybernetika (Prague),22(4), 320–328. · Zbl 0612.60035
[27] Michálek, J. (1986). Spectral decomposition of locally stationary random processes,Kybernetika (Prague),22(3), 244–255. · Zbl 0601.60036
[28] Michálek, J. (1989). Linear transformations of locally stationary processes,Apl. Mat.,34(1), 57–66. · Zbl 0672.60043
[29] Pesquet-Popescu, B. (1998).Modélisation bidimensionnelle de processus non-stationnaires et application à l’étude du fond sous-marin. Ph.D. thesis, Ecole Normale Supérieure de Cachan, July.
[30] Pesquet-Popescu, B. (1999). Wavelet packet analysis of 2d processes with stationary fractional increments,IEEE Trans. Inform. Theory, 1033–1039. · Zbl 0947.94002
[31] Pesquet-Popescu, B. and Larzabal, P. (1997). 2d-self-similar processes with stationary fractional increments. InFractals in Engineering, Tricot Levy Véhel, Lutton, Ed., Springer-Verlag, Berlin.
[32] Picinbono, B. (1974). Properties and applications of stochastic processes with stationarynth-order increments,Adv. Appl. Prob.,6, 512–523. · Zbl 0292.60067 · doi:10.2307/1426231
[33] Pinsker, M.S. (1955). The theory of curves in hilbert space with stationarynth increments, (Russian),Izv. Akad. Nauk SSSR. Ser. Mat.,19, 319–344.
[34] Pinsker, M.S. and Yaglom, A.M. (1954). Random processes with stationary increments of thenth order,Dokl. Akad. Nauk. SSSR,94, 385–388.
[35] Ramanathan, J. and Zeitouni, O. (1991). On the wavelet transform of fractional brownian motion,IEEE on Information Theory,37(4), 1156–1158. · doi:10.1109/18.87007
[36] Rozanov, Yu.A. (1959). Spectral analysis of abstract functions,Theor. Probab. Appl. · Zbl 0089.32602
[37] Schwartz, L. (1950).Théorie des Distributions, Hermann, Paris. · Zbl 0037.07301
[38] Silverman, R.A. (1957).Locally Stationary Random Processes. Div. Electromag. Res., Inst. Math. Sci., New York University. Res. Rep. No. MME-2.
[39] Tewfik, A.H. and Kim, M. (1992). Correlation structure of the discrete wavelet coefficients of fractional brownian motion,IEEE Trans. Inform. Theory,38(2), 904–909. · Zbl 0743.60079 · doi:10.1109/18.119750
[40] Veitch, D. and Abry, P. (1998). A wavelet based joint estimator for the parameters of Ird.,IEEE Trans. Info. Th., special issue ”Multiscale Statistical Signal Analysis and its Application”. · Zbl 0905.94006
[41] Winkler, H. (1993). Integral representation for stochastic processes withnth stationary increments,Math. Nachr.,163, 35–44. · Zbl 0801.60063 · doi:10.1002/mana.19931630105
[42] Wong, P.W. (1993). Wavelet decomposition of harmonizable random processes,IEEE Trans. Inform. Theory,39(1), 7–18. · Zbl 0768.60037 · doi:10.1109/18.179337
[43] Yaglom, A.M. (1958). Correlation theory of processes with randomnth increments,Am. Math. Soc. Transl.,8(2), 87–141. · Zbl 0080.34903
[44] Yaglom, A.M. (1987).Correlation Theory of Stationary and Related Random Functions, Vol. 1, Springer-Verlag, New York. · Zbl 0685.62077
[45] Yaglom, A.M. and Pinsker, M.S. (1953). Random processes with stationary increments of thenth order,Dokl. Akad. Nauk. SSSR,90, 731–734.
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