Skip to main content
SpringerLink
Log in

Higher-Order Spectral Densities of Fractional Random Fields

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

This paper presents the second- and higher-order spectral densities of stationary (in space) random fields arising as approximations of rescaled solutions of the heat and fractional heat equations with singular initial conditions. The development is based on the diagram formalism and the Riesz composition formula. Our results are the first step to full parametrization of higher-order spectra of some classes of fractional random fields.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. V. V. Anh, J. M. Angulo, and M. D. Ruiz-Medina, Possible long-range dependence in fractional random fields, J. Statist. Plann. Inference 80:95-110 (1999).

    Google Scholar 

  2. V. V. Anh, C. C. Heyde, and Q. Tieng, Stochastic models for fractal processes, J. Statist. Plann. Inference 80:123-135 (1999).

    Google Scholar 

  3. V. V. Anh and N. N. Leonenko, Non-Gaussian scenarios for the heat equation with singular initial data, Stochastic Process. Appl. 84:91-114 (1999).

    Google Scholar 

  4. V. V. Anh and N. N. Leonenko, Scaling laws for fractional diffusion-wave equation with singular initial data, Statist. Probab. Lett. 48:239-252 (2000).

    Google Scholar 

  5. V. V. Anh and N. N. Leonenko, Spectral analysis of fractional kinetic equations with random data, J. Statist. Phys. 104:1349-1387 (2001).

    Google Scholar 

  6. V. V. Anh and N. N. Leonenko, Renormalization and homogenization of fractional diffusion equations with random data, Probab. Theory Related Fields, (2002), in press.

  7. V. V. Anh, N. N. Leonenko, and L. M. Sakhno, Higher-order spectral densities of fractional random fields, Mathematical Report 07/02, Centre in Statistical Science and Industrial Mathematics, QUT, (2002).

  8. O. E. Barndorff-Nielsen and K. Prause, Apparent scaling, Finan. Stoch. 5:103-113 (2001).

    Google Scholar 

  9. O. E. Barndorff-Nielsen and N. Shephard, Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, part 2, J. Roy. Statist. Soc. Ser. B 63:167-241 (2001).

    Google Scholar 

  10. S. I. Boyarchenko and S. Z. Levendorskii, Non-Gaussian Merton-Black-Scholes theory, in Advanced Series on Statistical Science and Applied Probability, Vol. 9 (World Scientific, 2002).

  11. D. R. Brillinger, Introduction to polyspectra, Ann. Math. Statist. 36:1351-1374 (1965).

    Google Scholar 

  12. D. R. Brillinger, Fourier inference: Some methods for the analysis of array and nongaussian series data, Water Resources Bull. 21:743-756 (1985).

    Google Scholar 

  13. D. R. Brillinger and M. Rosenblatt, Asymptotic theory of estimates of kth order spectra, in Spectral Analysis of Time Series, B. Harris, ed. (Wiley, New York, 1967), pp. 153-188.

    Google Scholar 

  14. D. R. Brillinger and M. Rosenblatt, Computation and interpretation of kth order spectra, in Spectral Analysis of Time Series, B. Harris, ed. (Wiley, New York, 1967), pp. 189-232.

    Google Scholar 

  15. M. M. Djrbashian, Harmonic Analysis and Boundary Value Problems in Complex Domain (Birkhäuser Verlag, Basel, 1993).

    Google Scholar 

  16. R. L. Dobrushin, Gaussian and their subordinated self-similar random generalized fields, Ann. Probab. 7:1-28 (1979).

    Google Scholar 

  17. R. L. Dobrushin and P. Major, Non-central limit theorem for non-linear functionals of Gaussian fields, Z. Wahrsch. Verw. Gebiete 50:1-28 (1979).

    Google Scholar 

  18. N. du Plessis, An Introduction to Potential Theory (Oliver & Boyd, Edinburgh, 1970).

    Google Scholar 

  19. R. Fox and M. S. Taqqu, Multiple stochastic integrals with dependent integrators, J. Multivariate Anal. 21:105-127 (1987).

    Google Scholar 

  20. U. Frisch, Turbulence (Cambridge University Press, 1995).

  21. R. Hilfer, Fractional time evolution, in Applications of Fractional Calculus in Physics, R. Hilfer, ed. (World Scientific, Singapore, 2000), pp. 87-130.

    Google Scholar 

  22. A. V. Ivanov and N. N. Leonenko, Statistical Analysis of Random Fields (Kluwer, Dordrecht, 1989).

    Google Scholar 

  23. S. Jaffard, The multifractal nature of Lévy processes, Probab. Theory Related Fields 114:207-227 (1999).

    Google Scholar 

  24. S. Jaffard, On the Frisch-Parisi conjecture, J. Math. Pures Appl. 79:525-552 (2000).

    Google Scholar 

  25. J. Kampé de Feriet, Random solutions of the partial differential equations, in Proc. 3rd Berkeley Symp. Math. Stat. Prob., Volume III (University of California Press, Berkeley, CA, 1955), pp. 199-208.

    Google Scholar 

  26. A. N. Kochubei, Fractional order diffusion, J. Differential Equations 26:485-492 (1990).

    Google Scholar 

  27. N. N. Leonenko and W. Woyczynski, Scaling limits of solution of the heat equation with non-Gaussian data, J. Statist. Phys. 91:423-428 (1998).

    Google Scholar 

  28. F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett. 9:23-28 (1996).

    Google Scholar 

  29. F. Mainardi and R. Gorenflo, On Mittag-Leffler type functions in fractional evolution processes, J. Comput. Appl. Math. 118:283-299 (2000).

    Google Scholar 

  30. F. Mainardi, Y. Luchko, and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and Applied Analysis 4:153-192 (2001).

    Google Scholar 

  31. R. Metzler and J. Krafter, The random walk's guide to anomalous diffusion: a fractional dynamic approach, Phys. Rep. 339:1-72 (2000).

    Google Scholar 

  32. I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999).

    Google Scholar 

  33. M. B. Priestley, Non-linear and Non-stationary Time Series Analysis (Academic Press, London, 1988).

    Google Scholar 

  34. R. H. Riedi, Multifractal Processes, Technical Report TR99-06 (ECE Dept., Rice University, 1999).

  35. M. Rosenblatt, Remark on the Burgers equation, J. Math. Phys. 9:1129-1136 (1968).

    Google Scholar 

  36. M. Rosenblatt, Stationary Sequences and Random Fields (Birkhäuser, Boston, 1985).

    Google Scholar 

  37. W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30:134-144 (1989).

    Google Scholar 

  38. T. Subba Rao, On the theory of bilinear models, J. Roy. Statist. Soc. Ser. B 43:244-255 (1981).

    Google Scholar 

  39. T. Subba Rao and M. M. Gabr, An Introduction to Bispectral Analysis and Bilinear Time Series Models (Springer, Berlin, 1984).

    Google Scholar 

  40. M. S. Taqqu, Law of the iterated logarithm for sums of non-linear functions of Gaussian that exhibit a long-range dependence, Z. Wahrsch. Verw. Gebiete 40:203-238 (1977).

    Google Scholar 

  41. M. S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrsch. Verw. Gebiete 50:53-83 (1979).

    Google Scholar 

  42. G. Terdik, Bilinear Stochastic Models and related problems of nonlinear time series analysis, in Lecture Notes in Statistics, Vol. 142 (Springer-Verlag, New York, 1999).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, QLD, 4001, Australia

    V. V. Anh

  2. School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4YH, United Kingdom

    N. N. Leonenko

  3. Department of Mathematics, Kyiv University (National), Kyiv, 01033, Ukraine

    L. M. Sakhno

Authors
  1. V. V. Anh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. N. Leonenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. M. Sakhno
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Anh, V.V., Leonenko, N.N. & Sakhno, L.M. Higher-Order Spectral Densities of Fractional Random Fields. Journal of Statistical Physics 111, 789–814 (2003). https://doi.org/10.1023/A:1022898131682

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022898131682

Search

Navigation