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Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 157264, 26 pages
http://dx.doi.org/10.1155/2010/157264
Review Article

Fractal Time Series—A Tutorial Review

School of Information Science & Technology, East China Normal University, No. 500, Dong-Chuan Road, Shanghai 200241, China

Received 23 September 2009; Accepted 29 October 2009

Academic Editor: Massimo Scalia

Copyright © 2010 Ming Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. H. Griffel, Applied Functional Analysis, Ellis Horwood Series in Mathematics and Its Applications, John Wiley & Sons, New York, NY, USA, 1981. View at Zentralblatt MATH · View at MathSciNet
  2. C. K. Liu, Applied Functional Analysis, Defence Industry Press, Beijing, China, 1986.
  3. B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, San Francisco, Calif, USA, 1982. View at Zentralblatt MATH · View at MathSciNet
  4. G. Korvin, Fractal Models in the Earth Science, Elsevier, New York, NY, USA, 1992.
  5. E. E. Peters, Fractal Market Analysis—Applying Chaos Theory to Investment and Economics, John Wiley & Sons, New York, NY, USA, 1994.
  6. J. B. Bassingthwaighte, L. S. Liebovitch, and B. J. West, Fractal Physiology, Oxford University Press, New York, NY, USA, 1994.
  7. W. A. Fuller, Introduction to Statistical Time Series, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1996. View at MathSciNet
  8. G. E. P. Box, G. M. Jenkins, and G. C. Reinsel, Time Series Analysis: Forecasting and Control, Prentice Hall, Englewood Cliffs, NJ, USA, 3rd edition, 1994. View at Zentralblatt MATH · View at MathSciNet
  9. S. K. Mitra and J. F. Kaiser, Handbook for Digital Signal Processing, John Wiley & Sons, New York, NY, USA, 1993.
  10. J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedure, John Wiley & Sons, New York, NY, USA, 3rd edition, 2000.
  11. B. B. Mandelbrot, Gaussian Self-Affinity and Fractals, Selected Works of Benoit B. Mandelbrot, Springer, New York, NY, USA, 2002. View at MathSciNet
  12. J. Beran, Statistics for Long-Memory Processes, vol. 61 of Monographs on Statistics and Applied Probability, Chapman & Hall, New York, NY, USA, 1994. View at MathSciNet
  13. M. Li and P. Borgnat, “Forward for the special issue on traffic modeling, its computations and applications,” to appear in Telecommunication Systems. View at Publisher · View at Google Scholar
  14. J. Levy-Vehel, E. Lutton, and C. Tricot, Eds., Fractals in Engineering, Springer, New York, NY, USA, 1997.
  15. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing, Cambridge University Press, Cambridge, UK, 2nd edition, 2002. View at Zentralblatt MATH · View at MathSciNet
  16. H. Y.-F. Lam, Analog and Digital Filters: Design and Realization, Prentice Hall, Englewood Cliffs, NJ, USA, 1979.
  17. A. Papoulis, The Fourier Integral and Its Applications, McGraw-Hill, New York, NY, USA, 1962. View at MathSciNet
  18. R. O. Harger, An Introduction to Digital Signal Processing with MATHCAD, PWS Publishing, Boston, Mass, USA, 1999.
  19. J. Van de Vegte, Fundamentals of Digital Signal Processing, Prentice Hall, Englewood Cliffs, NJ, USA, 2003.
  20. M. Li, “Comparative study of IIR notch filters for suppressing 60-Hz interference in electrocardiogram signals,” International Journal of Electronics and Computers, vol. 1, no. 1, pp. 7–18, 2009.
  21. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  22. M. D. Ortigueira, “Introduction to fractional linear systems—part 1: continuous-time case,” IEE Proceedings: Vision, Image and Signal Processing, vol. 147, no. 1, pp. 62–70, 2000. View at Publisher · View at Google Scholar · View at Scopus
  23. M. D. Ortigueira, “Introduction to fractional linear systems—part 2: discrete-time case,” IEE Proceedings: Vision, Image and Signal Processing, vol. 147, no. 1, pp. 71–78, 2000. View at Publisher · View at Google Scholar · View at Scopus
  24. M. D. Ortigueira, “An introduction to the fractional continuous-time linear systems: the 21st century systems,” IEEE Circuits and Systems Magazine, vol. 8, no. 3, pp. 19–26, 2008. View at Publisher · View at Google Scholar · View at Scopus
  25. Y. Q. Chen and K. L. Moore, “Discretization schemes for fractional-order differentiators and integrators,” IEEE Transactions on Circuits and Systems I, vol. 49, no. 3, pp. 363–367, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  26. B. M. Vinagre, Y. Q. Chen, and I. Petras, “Two direct Tustin discretization methods for fractional-order differentiator/integrator,” Journal of the Franklin Institute, vol. 340, no. 5, pp. 349–362, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. B. B. Mandelbrot and J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Review, vol. 10, pp. 422–437, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. T. Hida, Brownian Motion, vol. 11 of Applications of Mathematics, Springer, New York, NY, USA, 1980. View at MathSciNet
  29. I. M. Gelfand and K. Vilenkin, Generalized Functions, vol. 1, Academic Press, New York, NY, USA, 1964.
  30. J. Mikusinski, Operational Calculus, International Series of Monographs on Pure and Applied Mathematics, Vol. 8, Pergamon Press, New York, NY, USA, 1959. View at MathSciNet
  31. M. Li and C.-H. Chi, “A correlation-based computational model for synthesizing long-range dependent data,” Journal of the Franklin Institute, vol. 340, no. 6-7, pp. 503–514, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. M. D. Ortigueira, “Comments on “Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions”,” Applied Mathematical Modelling, vol. 33, no. 5, pp. 2534–2537, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  33. M. D. Ortigueira and A. G. Batista, “On the relation between the fractional Brownian motion and the fractional derivatives,” Physics Letters A, vol. 372, no. 7, pp. 958–968, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  34. M. D. Ortigueira and A. G. Batista, “A fractional linear system view of the fractional brownian motion,” Nonlinear Dynamics, vol. 38, no. 1–4, pp. 295–303, 2004. View at Publisher · View at Google Scholar · View at Scopus
  35. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  36. S. C. Lim, M. Li, and L. P. Teo, “Locally self-similar fractional oscillator processes,” Fluctuation and Noise Letters, vol. 7, no. 2, pp. L169–L179, 2007. View at Publisher · View at Google Scholar · View at Scopus
  37. S. C. Lim, M. Li, and L. P. Teo, “Langevin equation with two fractional orders,” Physics Letters A, vol. 372, no. 42, pp. 6309–6320, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  38. M. Li and S. C. Lim, “A rigorous derivation of power spectrum of fractional Gaussian noise,” Fluctuation and Noise Letters, vol. 6, no. 4, pp. C33–C36, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  39. M. Li and S. C. Lim, “Power spectrum of generalized Cauchy process,” to appear in Telecommunication Systems. View at Publisher · View at Google Scholar
  40. B. B. Mandelbrot, Multifractals and 1/f Noise, Selected Works of B. B. Mandelbrot, Springer, New York, NY, USA, 1999. View at MathSciNet
  41. M. Li, “Fractional Gaussian noise and network traffic modeling,” in Proceedings of the 8th International Conference on Applied Computer and Applied Computational Science (WSEAS '09), pp. 34–39, Hangzhou, China, May 2009.
  42. M. Li, “Self-similarity and long-range dependence in teletraffic,” in Proceedings of the 9th International Conference on Applied Computer and Applied Computational Science (WSEAS '09), pp. 19–24, Hangzhou, China, May 2009.
  43. P. Abry, P. Borgnat, F. Ricciato, A. Scherrer, and D. Veitch, “Revisiting an old friend: on the observability of the relation between long range dependence and heavy tail,” to appear in Telecommunication Systems. View at Publisher · View at Google Scholar
  44. M. Li and S. C. Lim, “Modeling network traffic using generalized Cauchy process,” Physica A, vol. 387, no. 11, pp. 2584–2594, 2008. View at Publisher · View at Google Scholar · View at Scopus
  45. J. T. Kent and A. T. Wood, “Estimating the fractal dimension of a locally self-similar Gaussian process by using increments,” Journal of the Royal Statistical Society B, vol. 59, no. 3, pp. 679–699, 1997. View at Zentralblatt MATH · View at MathSciNet
  46. P. Hall and R. Roy, “On the relationship between fractal dimension and fractal index for stationary stochastic processes,” The Annals of Applied Probability, vol. 4, no. 1, pp. 241–253, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  47. A. J. Adler, The Geometry of Random Fields, John Wiley & Sons, New York, NY, USA, 1981. View at MathSciNet
  48. T. Gneiting and M. Schlather, “Stochastic models that separate fractal dimension and the Hurst effect,” SIAM Review, vol. 46, no. 2, pp. 269–282, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  49. S. C. Lim and M. Li, “A generalized Cauchy process and its application to relaxation phenomena,” Journal of Physics A, vol. 39, no. 12, pp. 2935–2951, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  50. M. S. Taqqu, V. Teverovsky, and W. Willinger, “Estimators for long-range dependence: an empirical study,” Fractals, vol. 3, no. 4, pp. 785–798, 1995.
  51. M. Li and W. Zhao, “Detection of variations of local irregularity of traffic under DDOS flood attack,” Mathematical Problems in Engineering, vol. 2008, Article ID 475878, 11 pages, 2008. View at Publisher · View at Google Scholar
  52. M. Li, “An approach to reliably identifying signs of DDOS flood attacks based on LRD traffic pattern recognition,” Computers & Security, vol. 23, no. 7, pp. 549–558, 2004. View at Publisher · View at Google Scholar · View at Scopus
  53. G. M. Raymond, D. B. Percival, and J. B. Bassingthwaighte, “The spectra and periodograms of anti-correlated discrete fractional Gaussian noise,” Physica A, vol. 322, no. 1–4, pp. 169–179, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  54. J. Beran, “Fitting long-memory models by generalized linear regression,” Biometrika, vol. 80, no. 4, pp. 817–822, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  55. J. Beran, “On parameter estimation for locally stationary long-memory processes,” Journal of Statistical Planning and Inference, vol. 139, no. 3, pp. 900–915, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  56. M. J. Cannon, D. B. Percival, D. C. Caccia, G. M. Raymond, and J. B. Bassingthwaighte, “Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series,” Physica A, vol. 241, no. 3-4, pp. 606–626, 1997. View at Publisher · View at Google Scholar · View at Scopus
  57. M. J. Cannon, D. B. Percival, D. C. Caccia, G. M. Raymond, and J. B. Bassingthwaighte, “Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series,” Physica A, vol. 241, no. 3-4, pp. 606–626, 1997. View at Publisher · View at Google Scholar · View at Scopus
  58. J. B. Bassingthwaighte and G. M. Raymond, “Evaluating rescaled range analysis for time series,” Annals of Biomedical Engineering, vol. 22, no. 4, pp. 432–444, 1994. View at Publisher · View at Google Scholar · View at Scopus
  59. S. E. Schepers, J. H. G. M. van Beek, and J. B. Bassingthwaighte, “Four methods to estimate the fractal dimension from self-affine signals,” IEEE Engineering in Medicine and Biology Magazine, vol. 11, no. 2, pp. 57–64, 1992.
  60. J. Mielniczuk and P. Wojdyłło, “Estimation of Hurst exponent revisited,” Computational Statistics & Data Analysis, vol. 51, no. 9, pp. 4510–4525, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  61. D. O. Cajueiro and B. M. Tabak, “The rescaled variance statistic and the determination of the Hurst exponent,” Mathematics and Computers in Simulation, vol. 70, no. 3, pp. 172–179, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  62. G. M. Raymond and J. B. Bassingthwaighte, “Deriving dispersional and scaled windowed variance analyses using the correlation function of discrete fractional Gaussian noise,” Physica A, vol. 265, no. 1-2, pp. 85–96, 1999. View at Publisher · View at Google Scholar · View at Scopus
  63. J. B. Bassingthwaighte and G. M. Raymond, “Evaluation of the dispersional analysis method for fractal time series,” Annals of Biomedical Engineering, vol. 23, no. 4, pp. 491–505, 1995.
  64. C. M. Kendziorski, J. B. Bassingthwaighte, and P. J. Tonellato, “Evaluating maximum likelihood estimation methods to determine the Hurst coefficient,” Physica A, vol. 273, no. 3-4, pp. 439–451, 1999. View at Publisher · View at Google Scholar · View at Scopus
  65. A. Guerrero and L. A. Smith, “A maximum likelihood estimator for long-range persistence,” Physica A, vol. 355, no. 2–4, pp. 619–632, 2005. View at Publisher · View at Google Scholar · View at Scopus
  66. D. Veitch and P. Abry, “A wavelet-based joint estimator of the parameters of long-range dependence,” IEEE Transactions on Information Theory, vol. 45, no. 3, pp. 878–897, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  67. C. Cattani, “Shannon wavelets theory,” Mathematical Problems in Engineering, vol. 2008, Article ID 164808, 24 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  68. C. Cattani, “Harmonic wavelet approximation of random, fractal and high frequency signals,” to appear in Telecommunication Systems.
  69. C. Cattani, “Harmonic wavelet analysis of a localized fractal,” International Journal of Engineering and Interdisciplinary Mathematics, vol. 1, no. 1, pp. 35–44, 2009.
  70. E. G. Bakhoum and C. Toma, “Relativistic short range phenomena and space-time aspects of pulse measurements,” Mathematical Problems in Engineering, vol. 2008, Article ID 410156, 20 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  71. G. W. Wornell, “Wavelet-based representations for the 1/f family of fractal processes,” Proceedings of the IEEE, vol. 81, no. 10, pp. 1428–1450, 1993. View at Publisher · View at Google Scholar · View at Scopus
  72. P. Abry, D. Veitch, and P. Flandrin, “Long-range dependence: revisiting aggregation with wavelets,” Journal of Time Series Analysis, vol. 19, no. 3, pp. 253–266, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  73. Y.-Q. Chen, R. Sun, and A. Zhou, “An improved Hurst parameter estimator based on fractional Fourier transform,” to appear in Telecommunication Systems. View at Publisher · View at Google Scholar
  74. R. B. Govindan, J. D. Wilson, H. Preißl, H. Eswaran, J. Q. Campbell, and C. L. Lowery, “Detrended fluctuation analysis of short datasets: an application to fetal cardiac data,” Physica D, vol. 226, no. 1, pp. 23–31, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  75. G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Modeling, Chapman & Hall, New York, NY, USA, 1994. View at MathSciNet
  76. T. G. Sinai, “Distribution of the maximum of fractional Brownian motion,” Russian Mathematical Surveys, vol. 52, no. 2, pp. 119–138, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  77. T. G. Sinai, “Self-similar probability distributions,” Theory of Probability & Its Applications, vol. 21, no. 1, pp. 63–80, 1976. View at Zentralblatt MATH · View at MathSciNet
  78. V. M. Sithi and S. C. Lim, “On the spectra of Riemann-Liouville fractional Brownian motion,” Journal of Physics A, vol. 28, no. 11, pp. 2995–3003, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  79. S. V. Muniandy and S. C. Lim, “Modeling of locally self-similar processes using multifractional Brownian motion of Riemann-Liouville type,” Physical Review E, vol. 63, no. 4, Article ID 046104, 7 pages, 2001. View at Publisher · View at Google Scholar · View at Scopus
  80. D. Feyel and A. de La Pradelle, “On fractional Brownian processes,” Potential Analysis, vol. 10, no. 3, pp. 273–288, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  81. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York, NY, USA, 1961.
  82. S. C. Lim and S. V. Muniandy, “On some possible generalizations of fractional Brownian motion,” Physics Letters A, vol. 266, no. 2-3, pp. 140–145, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  83. P. Flandrin, “On the spectrum of fractional Brownian motions,” IEEE Transactions on Information Theory, vol. 35, no. 1, pp. 197–199, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  84. R. F. Peltier and J. Levy-Vehel, “Multifractional Brownian motion: definition and preliminaries results,” INRIA TR 2645, 1995.
  85. R. F. Peltier and J. Levy-Vehel, “A new method for estimating the parameter of fractional Brownian motion,” INRIA TR 2696, 1994.
  86. M. Li, S. C. Lim, and W. Zhao, “Investigating multi-fractality of network traffic using local Hurst function,” Advanced Studies in Theoretical Physics, vol. 2, no. 10, pp. 479–490, 2008. View at Scopus
  87. S. V. Muniandy, S. C. Lim, and R. Murugan, “Inhomogeneous scaling behaviors in Malaysian foreign currency exchange rates,” Physica A, vol. 301, no. 1–4, pp. 407–428, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  88. M. Li, “Modeling autocorrelation functions of long-range dependent teletraffic series based on optimal approximation in Hilbert space—a further study,” Applied Mathematical Modelling, vol. 31, no. 3, pp. 625–631, 2007. View at Publisher · View at Google Scholar · View at Scopus
  89. J.-P. Chiles and P. Delfiner, Geostatistics, Modeling Spatial Uncertainty, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, 1999. View at MathSciNet
  90. P. Todorovic, An Introduction to Stochastic Processes and Their Applications, Springer Series in Statistics: Probability and Its Applications, Springer, New York, NY, USA, 1992. View at MathSciNet
  91. S. C. Lim and S. V. Muniandy, “Generalized Ornstein-Uhlenbeck processes and associated self-similar processes,” Journal of Physics A, vol. 36, no. 14, pp. 3961–3982, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  92. R. L. Wolpert and M. S. Taqqu, “Fractional Ornstein-Uhlenbeck Lévy processes and the telecom process: upstairs and downstairs,” Signal Processing, vol. 85, no. 8, pp. 1523–1545, 2005. View at Publisher · View at Google Scholar · View at Scopus
  93. M. Li and S. C. Lim, “Modeling network traffic using cauchy correlation model with long-range dependence,” Modern Physics Letters B, vol. 19, no. 17, pp. 829–840, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  94. M. Li and W. Zhao, “Representation of a stochastic traffic bound,” to appear in IEEE Transactions on Parallel and Distributed Systems.
  95. S. C. Lim and L. P. Teo, “Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure,” Stochastic Processes and Their Applications, vol. 119, no. 4, pp. 1325–1356, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  96. P. Vengadesh, S. V. Muniandy, and W. H. Abd.Majid, “Fractal morphological analysis of Bacteriorhodopsin (bR) layers deposited onto Indium Tin Oxide (ITO) electrodes,” Materials Science and Engineering C, vol. 29, no. 5, pp. 1621–1626, 2009. View at Publisher · View at Google Scholar · View at Scopus
  97. A. Scherrer, N. Larrieu, P. Owezarski, P. Borgnat, and P. Abry, “Non-Gaussian and long memory statistical characterizations for internet traffic with anomalies,” IEEE Transactions on Dependable and Secure Computing, vol. 4, no. 1, pp. 56–70, 2007. View at Publisher · View at Google Scholar · View at Scopus
  98. A. Karasaridis and D. Hatzinakos, “Network heavy traffic modeling using α-stable self-similar processes,” IEEE Transactions on Communications, vol. 49, no. 7, pp. 1203–1214, 2001. View at Publisher · View at Google Scholar · View at Scopus
  99. G. E. Uhlenbeck and L. S. Ornstein, “On the theory of the Brownian motion,” Physical Review, vol. 36, no. 5, pp. 823–841, 1930. View at Publisher · View at Google Scholar · View at Scopus
  100. D.-X. Lu, Stochastic Processes and Their Applications, Tsinghua University Press, Beijing, China, 2006.
  101. L. Valdivieso, W. Schoutens, and F. Tuerlinckx, “Maximum likelihood estimation in processes of Ornstein-Uhlenbeck type,” Statistical Inference for Stochastic Processes, vol. 12, no. 1, pp. 1–19, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  102. M. Shlesinger, G. M. Zaslavsky, and U. Frisch, Eds., Lévy Flights and Related Topics in Physics, Springer, New York, NY, USA, 1995.
  103. R. N. Mantegna and H. E. Stanley, “Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight,” Physical Review Letters, vol. 73, no. 22, pp. 2946–2949, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  104. J.-P. Bouchaud and A. Georges, “Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications,” Physics Reports, vol. 195, no. 4-5, pp. 127–293, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  105. D. Applebaum, “Lévy processes—from probability to finance and quantum groups,” Notices of the American Mathematical Society, vol. 51, no. 11, pp. 1336–1347, 2004. View at Zentralblatt MATH · View at MathSciNet
  106. R. J. Martin and A. M. Walker, “A power-law model and other models for long-range dependence,” Journal of Applied Probability, vol. 34, no. 3, pp. 657–670, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  107. R. J. Martin and J. A. Eccleston, “A new model for slowly-decaying correlations,” Statistics & Probability Letters, vol. 13, no. 2, pp. 139–145, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  108. M. Li, W. Jia, and W. Zhao, “Correlation form of timestamp increment sequences of self-similar traffic on Ethernet,” Electronics Letters, vol. 36, no. 19, pp. 1668–1669, 2000. View at Publisher · View at Google Scholar · View at Scopus
  109. F. Chapeau-Blondeau, “(max, +) dynamic systems for modeling traffic with long-range dependence,” Fractals, vol. 6, no. 4, pp. 305–311, 1998. View at Publisher · View at Google Scholar · View at Scopus
  110. B. Minasny and A. B. McBratney, “The Matérn function as a general model for soil variograms,” Geoderma, vol. 128, no. 3-4, pp. 192–207, 2005.
  111. W. Z. Daoud, J. D. W. Kahl, and J. K. Ghorai, “On the synoptic-scale Lagrangian autocorrelation function,” Journal of Applied Meteorology, vol. 42, no. 2, pp. 318–324, 2003. View at Publisher · View at Google Scholar · View at Scopus
  112. T. von Karman, “Progress in the statistical theory of turbulence,” Proceedings of the National Academy of Sciences of the United States of America, vol. 34, pp. 530–539, 1948. View at Zentralblatt MATH · View at MathSciNet