Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 4 (1999) > Paper 16 open journal systems 


Long-range Dependence trough Gamma-mixed Ornstein–Uhlenbeck Process

E. Iglói, L. Kossuth University
G. Terdik, L. Kossuth University


Abstract
The limit process of aggregational models---(i) sum of random coefficient AR(1) processes with independent Brownian motion (BM) inputs and (ii)  sum of AR(1) processes with random coefficients of Gamma distribution and with input of common BM's,---proves to be Gaussian and stationary and its transfer function  is the mixture of transfer functions of Ornstein--Uhlenbeck (OU) processes by Gamma distribution. It is called Gamma-mixed Ornstein--Uhlenbeck process  ($Gamma QTR{sf}{MOU}$QTR{sf}{)}. For independent  Poisson alternating $0$--$1$ reward processes with proper random intensity it is shown that the standardized sum of the processes converges to the standardized $Gamma QTR{sf}{MOU}$ process. The $Gamma QTR{sf}{MOU}$ process has various interesting properties and it is a new candidate for the successful modelling of several Gaussian stationary data with long-range dependence. Possible applications and problems are also considered.


Full text: PDF

Pages: 1-33

Published on: September 15, 1999
Bibliography
  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., April 1995, Vol. 3, 977–984, 1995.

    2. Math. Review number not available.
    (Abstract: http://www.sci.usq.edu.au/staff/addie/inf95abs.html)
  2. J. Bertoin, Sur une intégrale pour les processus à ?-variation bornée, Ann. Probab. 17(4), (1989), 1521–1535.

    3. MR91g:60065
  3. D. R. Brillinger and R. A. Irizarry, An investigation of the second- and higher-order spectra of music, Signal Process. 65(2), (1998), 161–179.

    4. Math. Review number not available.
    (Full article: http://stat-www.berkeley.edu/users/brill/papers.html)
  4. M. Buchanan, Fascinating rhythm, New Sci. 157(2115), (1998).

    5. Math. Review number not available.
    (Full article: http://www.newscientist.com/ns/980103/features.html)
  5. Ph. Carmona and L. Coutin, Fractional Brownian motion and the Markov property, Electron. Comm. Probab. 3, (1998), 95–107.

    6. MR1658690
  6. Ph. Carmona, L. Coutin and G. Montseny, Applications of a representation of long memory Gaussian processes. Submitted to Stochastic Process. Appl..

    7. Math. Review number not available.
    (Full article: http://www-sv.cict.fr/lsp/Carmona/prepublications.html)
  7. L. Coutin and L. Decreusefond, Stochastic differential equations driven by a fractional Brownian motion.

    8. Math. Review number not available.
    (Full article: http:// www.inf.enst.fr/~decreuse/recherche/recherche.html)
  8. D. R. Cox, Long-range dependence, non-linearity and time irreversibility, J. Time Ser. Anal. 12(4), (1991), 329–335.

    9. MR92f:62123
  9. W. Dai and C. C. Heyde, Itô's formula with respect to fractional Brownian motion and its application, J. Appl. Math. Stochastic Anal. 9(4), (1996), 439–448.

    10. MR98f:60104
  10. D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer-Verlag, New York, 1988.

    11. MR90e:60060
  11. L. Decreusefond and A. S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potential Anal. 10(2), (1999), 177–214.

    12. MR1677455
  12. C. W. J. Granger, Long memory relationships and the aggregation of dynamic models, J. Econometrics 14, (1980), 227–238.

    13. MR81m:62165
  13. J. M. Hausdorff and C.-K. Peng, Multi-scaled randomness: A possible source of 1/f noise in biology, Phys. Rev. E 54, (1996), 2154–2157.

    14. Math. Review number not available.
    (Full article:  http://reylab.bidmc.harvard.edu/publications/peng/1overf/1overf.html)
  14. E. Iglói and G. Terdik, Bilinear stochastic systems with fractional Brownian motion input, Ann. Appl. Probab. 9(1), (1999), 46–77.

    15. MR1682600
  15. F. Klingenhöfer and M. Zähle, Ordinary differential equations with fractal noise, Proc. Amer. Math. Soc. 127(4), (1999), 1021–1028.

    16. MR99f:60111
  16. 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), (1994), 1–15.

    17. Math. Review number not available.
    (Full article: http://www.acm.org/pubs/citations/journals/ton/1994-2-1/p1-leland/)
  17. S. B. Lowen and M. C. Teich, Fractal renewal processes generate 1/f noise, Phys. Rev. E 47, (1993), 992–1001.

    18. Math. Review number not available.
    (Full article: http://cordelia.mclean.org:8080/~lowen/pub_list.html)
  18. T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14(2), (1998), 215–310.

    19. MR1654527
  19. B. B. Mandelbrot, Long-run linearity, locally Gaussian processes, H-spectra and infinite variances, Internat. Econ. Rev. 10,- (1969), 82–113.

    20. Math. Review number not available.
  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, (1995), 82–87.

    21. Math. Review number not available.
    (Full article: http://reylab.bidmc.harvard.edu/publications/Chaos/phsg.html)
  21. B. Ryu and S. B. Lowen, Point process models for self-similar network traffic, with applications, Comm. Statist. Stochastic Models 14(3), (1998), 735–761.

    22. MR99a:60050
  22. 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(1--3), (1994), 214.

    23. Math. Review number not available.
    (Full article: http://ory.ph.biu.ac.il/~havlin/publications.txt.html)
  23. 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(3), (1993), 283.

    24. Math. Review number not available.
    (Full article: http://ory.ph.biu.ac.il/~havlin/publications.txt.html)
  24. H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probability 6(1), (1978), 19–41.

    25. MR57:1649
  25. M. S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, (1979), 53–83.

    26. MR81i:60020
  26. M. S. Taqqu, W. Willinger and R. Sherman, Proof of a fundamental result in self-similar traffic modeling, Computer Communication Review 27, (1997), 5–23.

    27. Math. Review number not available.
    (Full article: http://www.winlab.rutgers.edu/s3/reu/bgreading/bgreading.html )
  27. G. Terdik, Bilinear stochastic models and related problems of nonlinear time series analysis, a frequency domain approach, Lecture Notes in Statist. 142, Springer, New York, 1999.

    28. CMP 1 702 281
  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, (1995), 100–113.

    29. Math. Review number not available.
    (Full article: http://www.caida.org/Learn/tcpsrcmodel.html)
  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), (1997), 71–86.

    30. Math. Review number not available.
    (Full article: http://www.kric.ac.kr:8080/pubs/contents/journals/ton/1997-5/index.html)
  30. M. Zähle, Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Related Fields 111(3), (1998), 333–374.

    31. MR99j:60073













Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489