Methodology and Computing in Applied Probability

, Volume 7, Issue 3, pp 379–400 | Cite as

Simulation of Weakly Self-Similar Stationary Increment \({\text{Sub}}_{\varphi } {\left( \Omega \right)}\)-Processes: A Series Expansion Approach

  • Yuriy Kozachenko
  • Tommi SottinenEmail author
  • Olga Vasylyk


We consider simulation of \({\text{Sub}}_{\varphi } {\left( \Omega \right)}\)-processes that are weakly selfsimilar with stationary increments in the sense that they have the covariance function
$$R{\left( {t,s} \right)} = \frac{1}{2}{\left( {t^{{2H}} + s^{{2H}} - {\left| {t - s} \right|}^{{2H}} } \right)}$$
for some H ∈ (0, 1). This means that the second order structure of the processes is that of the fractional Brownian motion. Also, if \(H >\frac{1} {2}\) then the process is long-range dependent.

The simulation is based on a series expansion of the fractional Brownian motion due to Dzhaparidze and van Zanten. We prove an estimate of the accuracy of the simulation in the space C([0, 1]) of continuous functions equipped with the usual sup-norm. The result holds also for the fractional Brownian motion which may be considered as a special case of a \({\text{Sub}}_{{{x^{2} } \mathord{\left/ {\vphantom {{x^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} {\left( \Omega \right)}\)-process.


fractional Brownian motion φ-sub-Gaussian processes long-range dependence self-similarity series expansions simulation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. J. Beran, Statistics for Long-Memory Processes, Chapman and Hall: New York, 1994.Google Scholar
  2. V. V. Buldygin and Y. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, American Mathematical Society, Providence: RI, 2000.Google Scholar
  3. L. Decreusefond and A. S. Üstünel, “Stochastic analysis of the fractional Brownian motion,” Potential analysis vol. 10 (2) pp. 177–214, 1999.CrossRefGoogle Scholar
  4. P. Doukhan, G. Oppenheim, and M. Taqqu (eds.), Theory and applications of long-range dependence, Birkhäuser Boston, Inc.: Boston, MA, 2003.Google Scholar
  5. K. O. Dzhaparidze and J. H. van Zanten, “A series expansion of fractional Brownian motion,” Probability Theory and Related Fields vol. 130 pp. 39–55, 2004.CrossRefGoogle Scholar
  6. P. Embrechts and M. Maejima, Selfsimilar Processes, Princeton University Press: Princeton, 2002.Google Scholar
  7. G. A. Hunt, “Random Fourier transforms,” Transactions of the American Mathematical Society vol. 71 pp. 38–69, 1951.Google Scholar
  8. A. N. Kolmogorov, “Wienersche Spiralen und einige andere interessante Kurven in Hilbertschen Raum,” Comptes Rendus (Doklady) Acad. Sci. USSR (N.S.) vol. 26 pp. 115–118, 1940.Google Scholar
  9. Y. V. Kozachenko and O. I. Vasilik, “On the distribution of suprema of \({\text{Sub}}_{\varphi } {\left( \Omega \right)}\) random processes,” Theory of Stochastic Proc. vol. 4 (20) pp. 1–2, 1998, pp. 147–160.Google Scholar
  10. M. A. Krasnoselskii and Y. B. Rutitskii, Convex Functions in the Orlicz spaces, Fizmatiz: Moscow, 1958.Google Scholar
  11. B. Mandelbrot and J. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Review vol. 10 pp. 422–437, 1968.CrossRefGoogle Scholar
  12. G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian random processes, Chapman and Hall: New York, 1994.Google Scholar
  13. G. N. Watson, A Treatise of the Theory of Bessel Functions, Cambridge University Press: Cambridge, England, 1944.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Yuriy Kozachenko
    • 1
  • Tommi Sottinen
    • 2
    Email author
  • Olga Vasylyk
    • 1
  1. 1.Mechanics and Mathematics Faculty, Department of Probability Theory and Math. StatisticsTaras Shevchenko Kyiv National UniversityKyivUkraine
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

Personalised recommendations