Wavelet analysis and covariance structure of some classes of non-stationary processes

Journal of Fourier Analysis and Applications

Volumn 6, Issue 4, 2000, Pages 403-425

  Guérin, Charles Antoine ^a,b  

^a CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

^b AIX MARSEILLE UNIVERSITY   (France)

Author keywords

Fractional Brownian motion; Fractional increments; Locally stationary processes; Stationary increments; Stationary processes; Wavelet analysis

Indexed keywords

EID: 23044521625     PISSN: 10695869     EISSN: None     Source Type: Journal    
DOI: 10.1007/bf02510146     Document Type: Article

References (45)

1. Abry, P., Flandrin, P., Taqqu, M.S., and Veitch, D. (1999). Wavelets for the analysis, estimation and synthesis of scaling data, In Self-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.
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.
4. Brockwell, P.J. and Davis, R.A. (1987). Time Series: Theory and Methods, Springer-Verlag, New York.
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.
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.
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.
9. Doob, J.L. (1953). Stochastic Processes, Wiley Publications in Statistics, Wiley, New York.
10. Flandrin, P. (1989). On the spectrum of fractional brownian motion, IEEE Trans. Inform. Theory, 35(1), 197-199.
11. Flandrin, P. (1992). Wavelet analysis and synthesis of fractional brownian motion, IEEE Trans. Inform. Theory, 38(2), 910-917.
12. Folland, G.B. (1984). Real Analysis, John Wiley & Sons, New York.
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.
17. Hosking, J.R.M. (1981). Fractional differencing, Biometrika, 68(1), 165-176.
18. Houdré, C. (1990). Harmonizability, v-boundedness, (2,p)-boundedness of stochastic processes, Probab. Th. Rel. Fields, 87, 167-188.
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.
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.
23. Mandelbrot, B.B. and Van Ness, J.W. (1968). Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 422-437.
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.
25. Masry, E. (1996). Convergence properties of wavelet series expansions of fractional brownian motion, Appl. Comp. Harm. Anal., 3, 239-253.
26. Michálek, J. (1986). Ergodic properties of locally stationary processes, Kybernetika (Prague), 22(4), 320-328.
27. Michálek, J. (1986). Spectral decomposition of locally stationary random processes, Kybernetika (Prague), 22(3), 244-255.
28. Michálek, J. (1989). Linear transformations of locally stationary processes, Apl. Mat., 34(1), 57-66.
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.
31. Pesquet-Popescu, B. and Larzabal, P. (1997). 2d-self-similar processes with stationary fractional increments. In Fractals in Engineering, Tricot Levy Véhel, Lutton, Ed., Springer-Verlag, Berlin.
32. Picinbono, B. (1974). Properties and applications of stochastic processes with stationary nth-order increments, Adv. Appl. Prob., 6, 512-523.
33. Pinsker, M.S. (1955). The theory of curves in hilbert space with stationary nth 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 the nth 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.
36. Rozanov, Yu.A. (1959). Spectral analysis of abstract functions, Theor. Probab. Appl.
37. Schwartz, L. (1950). Théorie des Distributions, Hermann, Paris.
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.
40. Veitch, D. and Abry, P. (1998). A wavelet based joint estimator for the parameters of lrd., IEEE Trans. Info. Th., special issue "Multiscale Statistical Signal Analysis and its Application."
41. Winkler, H. (1993). Integral representation for stochastic processes with nth stationary increments, Math. Nachr., 163, 35-44.
42. Wong, P.W. (1993). Wavelet decomposition of harmonizable random processes, IEEE Trans. Inform. Theory, 39(1), 7-18.
43. Yaglom, A.M. (1958). Correlation theory of processes with random nth increments, Am. Math. Soc. Transl., 8(2), 87-141.
44. Yaglom, A.M. (1987). Correlation Theory of Stationary and Related Random Functions, Vol. 1, Springer-Verlag, New York.
45. Yaglom, A.M. and Pinsker, M.S. (1953). Random processes with stationary increments of the nth order, Dokl. Akad. Nauk. SSSR, 90, 731-734.