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Anisotropic scaling of the random grain model with application to network traffic

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  24 October 2016

Vytautė Pilipauskaitė  and
Donatas Surgailis
Vytautė Pilipauskaitė*
Affiliation:
Université de Nantes and Vilnius University
Donatas Surgailis*
Affiliation:
Vilnius University
*
* Postal address: Institute of Mathematics and Informatics, Akademijos 4, LT-08663 Vilnius, Lithuania.
* Postal address: Institute of Mathematics and Informatics, Akademijos 4, LT-08663 Vilnius, Lithuania.
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Abstract

We obtain a complete description of anisotropic scaling limits of the random grain model on the plane with heavy-tailed grain area distribution. The scaling limits have either independent or completely dependent increments along one or both coordinate axes and include stable, Gaussian, and ‘intermediate’ infinitely divisible random fields. The asymptotic form of the covariance function of the random grain model is obtained. Application to superimposed network traffic is included.

Keywords

Random grain modelanisotropic scalinglong-range dependenceLévy sheetfractional Brownian sheetworkload process

MSC classification

Primary: 60G60: Random fields
Secondary: 60G22: Fractional processes, including fractional Brownian motion 60G51: Processes with independent increments; L'evy processes 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 857 - 879
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1] Albeverio, S.,Molchanov, S. A. and Surgailis, D (1994).Stratified structure of the Universe and Burgers' equation - a probabilistic approach.Prob. Theory Relat. Fields 100,457–484.Google Scholar
[2] Anh, V. V.,Leonenko, N. N. and Ruiz-Medina, M. D. (2013).Macroscaling limit theorems for filtered spatiotemporal random fields.Stoch. Anal. Appl. 31,460–508.Google Scholar
[3] Biermé, H.,Estrade, A. and Kaj, I. (2010).Self-similar random fields and rescaled random balls models.J. Theoret. Prob. 23,1110–1141.CrossRefGoogle Scholar
[4] Biermé, H.,Meerschaert, M. M. and Scheffler, H.-P. (2007).Operator scaling stable random fields.Stoch. Process. Appl. 117,312–332.Google Scholar
[5] Breuer, P. and Major, P. (1983).Central limit theorems for nonlinear functionals of Gaussian fields.J. Multivariate Anal. 13,425–441.CrossRefGoogle Scholar
[6] Dobrushin, R. L. and Major, P. (1979).Non-central limit theorems for nonlinear functionals of Gaussian fields.Z. Wahrscheinlichkeitsth. 50,27–52.Google Scholar
[7] Dombry, . and Kaj, I. (2011).The on-off network traffic under intermediate scaling..Queueing Systems 69,29–44.Google Scholar
[8] Gaigalas, R. (2006).A Poisson bridge between fractional Brownian motion and stable Lévy motion.Stoch. Process. Appl. 116,447–462.CrossRefGoogle Scholar
[9] Gaigalas, R. and Kaj, I. (2003).Convergence of scaled renewal processes and a packet arrival model.Bernoulli 9,671–703.Google Scholar
[10] Kaj, I. and Taqqu, M. S. (2008).Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach. In In and Out of Equilibrium, 2, (Progr. Prob. 60),Birkhäuser,Basel, pp.383–427 Google Scholar
[11] Kaj, I.,Leskelä, L.,Norros, I. and Schmidt, V. (2007).Scaling limits for random fields with long-range dependence. Ann. Prob. 35,528–550.CrossRefGoogle Scholar
[12] Lavancier, F.(2007).Invariance principles for non-isotropic long memory random fields.Statist. Inf. Stoch. Process. 10,255–282.Google Scholar
[13] Leipus, R.,Paulauskas, V. and Surgailis, D. (2005).Renewal regime switching and stable limit laws.J. Econometrics 129,299–327.CrossRefGoogle Scholar
[14] Leonenko, N. N. and Olenko, A. (2013).Tauberian and Abelian theorems for long-range dependent random fields.Methodol. Comput. Appl. Prob. 15,715–742.CrossRefGoogle Scholar
[15] Lifshits, M. (2014).Random Processes by Example.World Scientific,Hackensack, NJ.Google Scholar
[16] Mikosch, T.,Resnick, S.,Rootzén, H. and Stegeman, A. (2002).Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Prob. 12,23–68.Google Scholar
[17] Pilipauskaitė, V. and Surgailis, D. (2014). Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes.Stoch. Process. Appl. 124,1011–1035.Google Scholar
[18] Pilipauskaitė, V. and Surgailis, D. (2015).Anisotropic scaling of random grain model with application to network traffic. Available at http://arxiv.org/abs/1510.07423.Google Scholar
[19] Pilipauskaitė, V. and Surgailis, D. (2015).Joint aggregation of random-coefficient AR(1) processes with common innovations.Statist. Prob. Lett. 101,73–82.Google Scholar
[20] Puplinskaitė, V. and Surgailis, D. (2015).Scaling transition for long-range dependent Gaussian random fields.Stoch. Process. Appl. 125,2256–2271.Google Scholar
[21] Puplinskaitė, D. and Surgailis, D. (2016).Aggregation of autoregressive random fields and anisotropic long-range dependence.Bernoulli 22,2401–2441.CrossRefGoogle Scholar
[22] Surgailis, D. (1982).Zones of attraction of self-similar multiple integrals.Lithuanian Math. J. 22,327–340.Google Scholar
[23] Wang, Y. (2014).An invariance principle for fractional Brownian sheets.J. Theoret. Prob. 27,1124–1139.Google Scholar