Long-range dependence through gamma-mixed Ornstein-Uhlenbeck process

Electronic Journal of Probability

Volumn 4, Issue , 1999, Pages 1-33

  Igloi E ^a   Terdik, G ^a  

^a UNIVERSITY OF DEBRECEN   (Hungary)

Author keywords

Aggregational model; Fractional Brownian motion input; Heart rate variability; Long range dependence; Ornstein Uhlenbeck process; Spectral representation; Stationarity; Stochastic differential equation

Indexed keywords

EID: 0002119933     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v4-53     Document Type: Article

References (30)

1. R. G. Addie, M. Zukerman, and T. Neame. Fractal traffic: Measurements, modelling and performance evaluation. In Proceedings of IEEE Infocom’95, Boston, MA, U.S.A., volume 3, pages 977-984, April 1995.
2. J. Bertoin. Sur une integrale pour les processus a a-variation bornee. Ann. Probab., 17(4):1521-1535, 1989.
3. D. R. Brillinger and R. A. Irizarry. An investigation of the second-and higher-order spectra of music. Signal Process., 65(2):161-179, 1998.
4. M. Buchanan. Fascinating rhythm. New Sci., 157(2115), 1998.
5. Ph. Carmona and L. Coutin. Fractional Brownian motion and the Markov property. Electronic Communications in Probability, 3:95-107, 1998.
6. Ph. Carmona, L. Coutin, and G. Montseny. Applications of a representation of long memory Gaussian processes. Submitted to Stochastic Process. Appl. (www-sv.cict.fr/lsp/Carmona/prepublications.html).
7. L. Coutin and L. Decreusefond. Stochastic differential equations driven by a fractional Brownian motion. To appear.
8. D. R. Cox. Long-range dependence, non-linearity and time irreversibility. J. Time Ser. Anal., 12(4):329-335, 1991.
9. W. Dai and C. C. Heyde. Ito’s formula with respect to fractional Brownian motion and its application. J. Appl. Math. Stochastic Anal., 9(4):439-448, 1996.
10. L. Decreusefond and A. S. Üstünel. Stochastic analysis of the fractional Brownian motion. Potential Anal., 10(2):177-214, 1999.
11. D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Springer-Verlag, New York, 1988.
12. C. Granger. Long-memory relationship and aggregation of dynamic models. Journ. of Econometrics, 14:227-238, 1980.
13. J. M. Hausdorff and C.-K. Peng. Multi-scaled randomness: a possible source of 1/f noise in biology. Physical Review E, 54:2154-2157, 1996.
14. E. Igloi and G. Terdik. Bilinear stochastic systems with fractional Brownian motion input. Ann. Appl. Probab., 9:46-77, 1999.
15. F. Klingenhofer and M. Zahle. Ordinary differential equations with fractal noise. Proc. Amer. Math. Soc., 127(4):1021-1028, 1999.
16. S. B. Lowen and M. C. Teich. Fractal renewal processes generate 1/f noise. Phys. Rev. E, 47:992-1001, 1993.
17. W. E. Leland, M. S. Taqqu, W. Willinger, and D. V. Wilson. On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Transactions on Networking, 2(1):1-15, 1994.
18. T. J. Lyons. Differential equations driven by rough signals. Revista Matemática Iberoamericana, 14(2):215-310, 1998.
19. B. B. Mandelbrot. Long-run linearity, locally Gaussian processes, H-spectra and infinite variances. Internat. Econ. Rev., 10:82-113, 1969.
20. C.-K. Peng, S. Havlin, H. E. Stanley, and A. L. Goldberger. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos, 5:82-87, 1995.
21. B. Ryu and S. B. Lowen. Point process models for self-similar network traffic, with applications. Comm. Statist. Stochastic Models, 14(3):735-761, 1998.
22. H. E. Stanley, S. V. Buldyrev, A. L. Goldberger, S. Havlin, S. M. Ossadnik, C.-K. Peng, and M. Simons. Fractal landscapes in biological systems. Fractals, 1:283, 1993.
23. H. E. Stanley, S. V. Buldyrev, A. L. Goldberger, Z. D. Goldberger, S. Havlin, S. M. Ossadnik, C.-K. Peng, and M. Simons. Statistical mechanics in biology: How ubiquitous are long-range correlations? Physica A, 205:214, 1994.
24. H. J. Sussmann. On the gap between deterministic and stochastic ordinary differential equations. Ann. Probability, 6(1):19-41, 1978.
25. M. S. Taqqu. Convergence of integrated processes of arbitrary Hermite rank. Zeitschrift fur Wahrscheinlichkeitstheorie Verw. Gebiete, 50:53-83, 1979.
26. Gy. Terdik. Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis; A Frequency Domain Approach, volume 142 of Lecture Notes in Statistics. Springer Verlag, New York, 1999.
27. M. S. Taqqu, W. Willinger, and R. Sherman. Proof of a fundamental result in self-similar traffic modeling. Computer Communication Review, 27:5-23, 1997.
28. W. Willinger, M. S. Taqqu, R. Sherman, and D. V. Wilson. Self-similarity through high variability: Statistical analysis of Ethernet LAN traffic at the source level. Computer Communication Review, 25:100-113, 1995.
29. W. Willinger, M. S. Taqqu, R. Sherman, and D. V. Wilson. Self-similarity through high-variability: Statistical analysis of Ethernet LAN traffic at the source level. IEEE/ACM Transactions on Networking, 5(1):1-16, 1997.
30. M. Zähle. Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields, 111(3):333-374, 1998.