Annals of Probability

Scaling limits for random fields with long-range dependence

Ingemar Kaj, Lasse Leskelä, Ilkka Norros, and Volker Schmidt

Full-text: Open access


This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density λ of the sets grows to infinity and the mean volume ρ of the sets tends to zero. Assuming that the volume distribution has a regularly varying tail with infinite variance, we show that the centered and renormalized random field can have three different limits, depending on the relative speed at which λ and ρ are scaled. If λ grows much faster than ρ shrinks, the limit is Gaussian with long-range dependence, while in the opposite case, the limit is independently scattered with infinite second moments. In a special intermediate scaling regime, there exists a nontrivial limiting random field that is not stable.

Article information

Ann. Probab., Volume 35, Number 2 (2007), 528-550.

First available in Project Euclid: 30 March 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G60: Random fields 60G18: Self-similar processes

Long-range dependence self-similar random field fractional Brownian motion fractional Gaussian noise stable random measure Riesz energy


Kaj, Ingemar; Leskelä, Lasse; Norros, Ilkka; Schmidt, Volker. Scaling limits for random fields with long-range dependence. Ann. Probab. 35 (2007), no. 2, 528--550. doi:10.1214/009117906000000700.

Export citation


  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
  • Gaigalas, R. and Kaj, I. (2003). Convergence of scaled renewal processes and a packet arrival model. Bernoulli 9 671–703.
  • Gripenberg, G. and Norros, I. (1996). On the prediction of fractional Brownian motion. J. Appl. Probab. 33 400–410.
  • Kaj, I. (2005). Limiting fractal random processes in heavy-tailed systems. In Fractals in Engineering, New Trends in Theory and Applications (J. Lévy-Léhel and E. Lutton, eds.) 199–218. Springer, London.
  • Kaj, I. and Martin-L öf, A. (2005). Scaling limit results for the sum of many inverse Lévy subordinators. Preprint, Institut Mittag–Leffler.
  • Kaj, I. and Taqqu, M. S. (2004). Convergence to fractional Brownian motion and to the Telecom process: The integral representation approach. Preprint, Dept. Mathematics, Uppsala Univ.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • Konstantopoulos, T. and Lin, S.-J. (1998). Macroscopic models for long-range dependent network traffic. Queueing Syst. 28 215–243.
  • Kurtz, T. G. (1996). Limit theorems for workload input models. In Stochastic Networks: Theory and Applications (F. P. Kelly, S. Zachary and I. Ziedins, eds.) 119–139. Oxford Univ. Press.
  • Landkof, N. S. (1972). Foundations of Modern Potential Theory. Springer, New York.
  • Mandelbrot, B. B. and van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422–437.
  • Mikosch, T., Resnick, S. I., Rootzén, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Probab. 12 23–68.
  • Rudin, W. (1987). Real and Complex Analysis, 3rd ed. McGraw–Hill, New York.
  • Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastics Models with Infinite Variance. Chapman and Hall, London.
  • Taqqu, M. S. and Levy, J. B. (1986). Using renewal processes to generate long-range dependence and high variability. In Dependence in Probability and Statistics (E. Eberlein and M. S. Taqqu, eds.) 73–89. Birkhäuser, Boston.