Long Range Dependence in Third Order for Non-Gaussian Time Series



The object of this paper is to define the long-range dependence (LRD) for a Non-Gaussian time series in third order and to investigate the third order properties of some well known long-range dependent series. We define the third order LRD in terms of the third order cumulants and of the bispectrum. The definition of the third order LRD is given in polar coordinates.


Fractional Brownian Motion Hurst Exponent Order Property Range Dependence Order Cumulants 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amblard PO, Brossier JM, Lacoume JL (1997) Playing with long range dependence and HOS. Proceedings of the IEEE signal processing, workshop on higher-order statistics, 1997, IEEE Xplore, pp 453–457CrossRefGoogle Scholar
  2. 2.
    Beran J (1994) Statistics for long-memory processes. Monographs on statistics and applied probability, vol 61. Chapman and Hall, LondonGoogle Scholar
  3. 3.
    Bingham NH, Goldie CM, Teugels JL (1987) Regular variation. Encyclopedia of mathematics and its applications, vol 27. Cambridge University Press, CambridgeGoogle Scholar
  4. 4.
    Breuer P, Major P (1983) Central limit theorems for nonlinear functionals of Gaussian fields. J Multivar Anal 13(3):425–441CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Brillinger DR (1965) An introduction to polyspectra. Ann Math Stat 36:1351–1374CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Brillinger DR (1985) Fourier inference: Some methods for the analysis of array and nongaussian series data. Wat Resour Bull 21:743–756Google Scholar
  7. 7.
    Brillinger DR, Irizarry RA (1998) An investigation of the second- and higher-order spectra of music. Signal Process 65(2):161–179CrossRefMATHGoogle Scholar
  8. 8.
    Brillinger DR, Rosenblatt M (1967) Asymptotic theory of k-th order spectra. In: Harris B (ed) Spectral analysis of time series. Wiley, New York, pp 153–188Google Scholar
  9. 9.
    Dobrushin RL, Major P (1979) Non-central limit theorems for non-linear functionals of Gaussian fields. Z Wahrsch verw Gebiete 50:27–52CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Doukhan P (2003) Models, inequalities, and limit theorems for stationary sequences. Theory and applications of long-range dependence. Birkhäuser, Boston, pp 43–100Google Scholar
  11. 11.
    Giraitis L, Robinson PM (2003) Parametric estimation under long-range dependence. Theory and applications of long-range dependence. Birkhäuser, Boston, pp 229–249Google Scholar
  12. 12.
    Granger CWJ, Ding Z (1996) Varieties of long memory models. J Econom 73(1):61–77CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Granger CWJ, Joyeux R (1980) An introduction to long memory time series models and fractional differencing. J Time Ser Anal 1:15–30CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Hinich MJ (1982) Testing for Gaussianity and linearity of a stationary time series. J Time Ser Anal 3L169–176CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Hinich MJ, Messer H (1995) On the principal domain of the discrete bispectrum of a stationary signal. IEEE Trans Signal Process 43(9):2130–2134CrossRefGoogle Scholar
  16. 16.
    Hosking JRM (1981) Fractional differencing. Biometrika 68:165–167CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Hurst HE (1951) Long term storage capacity of reservoirs. Trans Am Soc Civil Eng 116: 770–808Google Scholar
  18. 18.
    Iglói E, Terdik Gy (2003) Superposition of diffusions with linear generator and its multifractal limit process. ESAIM Probab Stat 7:23–88 (electronic)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Jammalamadaka SR, Subba Rao T, Terdik Gy (2007) On multivariate nonlinear regression models with stationary correlated errors. J Stat Plan Inference R S N 137:3793–3814CrossRefMATHGoogle Scholar
  20. 20.
    Konovalov SP (1979) Absolute convergence of multiple Fourier series. Mat Zametki 25(2):211–216, 317MathSciNetGoogle Scholar
  21. 21.
    Major P (1981) Multiple Wiener–Itô integrals. Lecture notes in mathematics, vol 849. Springer, New YorkMATHGoogle Scholar
  22. 22.
    Molnár S, Terdik Gy (2001) A general fractal model of internet traffic. In: Proceedengs of the 26th annual IEEE conference on local computer networks (LCN), Tampa, Florida, USA (Los Alamitos, CA, USA), IEEE Computer Society, IEEE, November 2001, pp 492–499Google Scholar
  23. 23.
    Rosenblatt M (1961) Independence and dependence. In: Proceedings of 4th Berkeley symposium on Mathematical Statistics and Probability, vol II. University of California Press, California, pp 431–443Google Scholar
  24. 24.
    Rosenblatt M (1976) Fractional integrals of stationary processes and the central limit theorem. J Appl Probab 13(4):723–732CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Rosenblatt M (1979) Some limit theorems for partial sums of quadratic forms in stationary Gaussian variables. Z Wahrsch Verw Gebiete 49(2):125–132CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Rosenblatt M (1981) Limit theorems for Fourier transforms of functionals of Gaussian sequences. Z Wahrsch Verw Gebiete 55(2):123–132CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Rosenblatt M (1987) Remarks on limit theorems for nonlinear functionals of Gaussian sequences. Probab Theory Relat Fields 75(1):1–10CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Samorodnitsky G, Taqqu MS (1994) Stable non-Gaussian random processes. Stochastic modeling, Stochastic models with infinite variance. Chapman and Hall, New YorkGoogle Scholar
  29. 29.
    Scherrer A, Larrieu N, Owezarski P, Borgnat P, Abry P (2007) Non-gaussian and long memory statistical characterizations for internet traffic with anomalies. IEEE Trans Dependable Secur Comput 4(1):56–70CrossRefGoogle Scholar
  30. 30.
    Stein EM, Weiss G (1971) Introduction to Fourier analysis on Euclidean spaces. Princeton mathematical series, vol 32. Princeton University Press, PrincetonGoogle Scholar
  31. 31.
    Subba Rao T, Gabr MM (1984) An introduction to bispectral analysis and Bilinear Time series. Lecture notes in statistics, vol 24. Springer, New YorkMATHGoogle Scholar
  32. 32.
    Taqqu MS (1975) Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z Wahrsch verw Gebiete 31:287–302CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Taqqu MS (1979) Convergence of integrated processes of arbitrary Hermite rank. Z Wahrsch Verw Gebiete 50:53–83CrossRefMathSciNetMATHGoogle Scholar
  34. 34.
    Taqqu MS (2003) Fractional Brownian motion and long-range dependence. Theory and applications of long-range dependence. Birkhäuser, Boston, pp 5–38Google Scholar
  35. 35.
    Terdik Gy (1999) Bilinear stochastic models and related problems of nonlinear time series analysis; a frequency domain approach. Lecture notes in statistics, vol 142. Springer, New YorkGoogle Scholar
  36. 36.
    Terdik Gy (2008) Long-range dependence in higher order for non-gaussian time series. Acta Sci Math (Szeged) 74:663–684Google Scholar
  37. 37.
    Terdik Gy, Gyires T (2009) Lévy flights and fractal modeling of internet traffic. IEEE/ACM Trans Netw 17(1):120–129CrossRefGoogle Scholar
  38. 38.
    Wainger S (1965) Special trigonometric series in k-dimensions. Mem Am Math Soc 59:102MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of Information Technology, Faculty of InformaticsUniversity of DebrecenDebrecenHungary

Personalised recommendations