Advances in Applied Probability

On long-range dependence of random measures

Daniel Vašata

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper deals with long-range dependence of random measures on ℝd. By examples, it is demonstrated that one must be careful in order to define it consistently. Therefore, we define long-range dependence by a rather specific second-order condition and provide an equivalent formulation involving the asymptotic behaviour of the Bartlett spectrum near the origin. Then it is shown that the defining condition may be formulated less strictly when the additional isotropy assumption holds. Finally, we present an example of a long-range dependent random measure based on the 0-level excursion set of a Gaussian random field for which the corresponding spectral density and its asymptotics are explicitly derived.

Article information

Adv. in Appl. Probab., Volume 48, Number 4 (2016), 1235-1255.

First available in Project Euclid: 24 December 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A40: Inequalities and extremum problems

Long-range dependence random measure Bartlett spectrum


Vašata, Daniel. On long-range dependence of random measures. Adv. in Appl. Probab. 48 (2016), no. 4, 1235--1255.

Export citation


  • Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
  • Beran, J., Feng, Y., Ghosh, S. and Kulik, R. (2013). Long-Memory Processes: Probabilistic Properties and Statistical Methods. Springer, Heidelberg.
  • Berg, C. and Forst, G. (1975). Potential Theory on Locally Compact Abelian Groups. Springer, New York.
  • Brandolini, L., Hofmann, S. and Iosevich, A. (2003). Sharp rate of average decay of the Fourier transform of a bounded set. Geom. Funct. Anal. 13, 671–680.
  • Chiu, S. N., Stoyan, D., Kendall, W. S. and Mecke, J. (2013). Stochastic Geometry and Its Applications, 3rd edn. John Wiley, Chichester.
  • Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.
  • Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, 2nd edn. Springer, New York.
  • Daley, D. J. and Vesilo, R. (1997). Long range dependence of point processes, with queueing examples. Stoch. Process. Appl. 70, 265–282.
  • Donoghue, W. F., Jr. (1969). Distributions and Fourier Transforms. Academic Press, New York.
  • Gneiting, T. and Schlather, M. (2004). Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev. 46, 269–282.
  • Iosevich, A. and Liflyand, E. (2014). Decay of the Fourier Transform. Birkhäuser, Basel.
  • Krasikov, I. (2006). Uniform bounds for Bessel functions. J. Appl. Anal. 12, 83–91.
  • Kuronen, M. and Leskelä, L. (2013). Hard-core thinnings of germ–grain models with power-law grain sizes. Adv. Appl. Prob. 45, 595–625.
  • Reed, M. and Simon, B. (1975). Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press, New York.
  • Samorodnitsky, G. (2006). Long range dependence. Foundations Trends Stoch. Systems 1, 163–257.
  • Schneider, R. (2014). Convex Bodies: The Brunn–Minkowski Theory, 2nd edn. Cambridge University Press.
  • Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
  • Stein, E. M. and Weiss, G. (1971). Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press.
  • Temme, N. M. (1996). Special Functions. John Wiley, New York.
  • Watson, G. N. (1995). A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press.