Long-range Dependence trough Gamma-mixedOrnstein-Uhlenbeck Process

E. Igloi (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 ($\Gamma\mathsf{MOU}$). 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 $\Gamma\mathsf{MOU}$ process. The $\Gamma\mathsf{MOU}$ 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: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-33

Publication Date: September 15, 1999

DOI: 10.1214/EJP.v4-53

References

  • 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.
  • Bertoin, Jean. Sur une intégrale pour les processus à $\alpha$-variation bornée. (French) [On an integral for processes with bounded $\alpha$-variation] Ann. Probab. 17 (1989), no. 4, 1521--1535. MR1048943
  • 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.
  • M. Buchanan. Fascinating rhythm. New Sci., 157(2115), 1998.
  • Carmona, Philippe; Coutin, Laure. Fractional Brownian motion and the Markov property. Electron. Comm. Probab. 3 (1998), 95--107 (electronic). MR1658690
  • 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).
  • L. Coutin and L. Decreusefond. Stochastic differential equations driven by a fractional Brownian motion. To appear.
  • Cox, D. R. Long-range dependence, nonlinearity and time irreversibility. J. Time Ser. Anal. 12 (1991), no. 4, 329--335. MR1131005
  • Dai, W.; Heyde, C. C. Itô's formula with respect to fractional Brownian motion and its application. J. Appl. Math. Stochastic Anal. 9 (1996), no. 4, 439--448. MR1429266
  • Decreusefond, L.; Üstünel, A. S. Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1999), no. 2, 177--214. MR1677455
  • Daley, D. J.; Vere-Jones, D. An introduction to the theory of point processes. Springer Series in Statistics. Springer-Verlag, New York, 1988. xxii+702 pp. ISBN: 0-387-96666-8 MR0950166
  • Granger, C. W. J. Long memory relationships and the aggregation of dynamic models. J. Econometrics 14 (1980), no. 2, 227--238. MR0597259
  • 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.
  • Iglói, E.; Terdik, Gy. Bilinear stochastic systems with fractional Brownian motion input. Ann. Appl. Probab. 9 (1999), no. 1, 46--77. MR1682600
  • Klingenhöfer, F.; Zähle, M. Ordinary differential equations with fractal noise. Proc. Amer. Math. Soc. 127 (1999), no. 4, 1021--1028. MR1486738
  • S. B. Lowen and M. C. Teich. Fractal renewal processes generate 1/f noise. Phys. Rev. E, 47:992--1001, 1993.
  • 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.
  • Lyons, Terry J. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), no. 2, 215--310. MR1654527
  • B. B. Mandelbrot. Long-run linearity, locally Gaussian processes, H-spectra and infinite variances. Internat. Econ. Rev., 10:82--113, 1969.
  • 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.
  • Ryu, Bo; Lowen, Steven B. Point process models for self-similar network traffic, with applications. Comm. Statist. Stochastic Models 14 (1998), no. 3, 735--761. MR1621303
  • 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.
  • 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.
  • Sussmann, Héctor J. On the gap between deterministic and stochastic ordinary differential equations. Ann. Probability 6 (1978), no. 1, 19--41. MR0461664
  • Taqqu, Murad S. Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 53--83. MR0550123
  • Terdik, György. Bilinear stochastic models and related problems of nonlinear time series analysis. A frequency domain approach. Lecture Notes in Statistics, 142. Springer-Verlag, New York, 1999. xx+260 pp. ISBN: 0-387-98872-6 MR1702281
  • 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.
  • 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.
  • 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.
  • Zähle, M. Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields 111 (1998), no. 3, 333--374. MR1640795
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.