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On long-range dependence of random measures

Part of: Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  11 January 2017

Daniel Vašata
Daniel Vašata*
Affiliation:
Czech Technical University in Prague
*
* Postal address: Department of Applied Mathematics, Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, Prague, CZ-160 00, Czech Republic. Email address: daniel.vasata@fit.cvut.cz
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Abstract

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.

Keywords

Long-range dependencerandom measureBartlett spectrum

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 1235 - 1255
Copyright
Copyright © Applied Probability Trust 2017 

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References

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