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Scaling limits of nonlinear functions of random grain model, with application to Burgers’ equation

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  • Surgailis, Donatas

Abstract

We study scaling limits of nonlinear functions G of random grain model X on Rd with long-range dependence and marginal Poisson distribution. Following Kaj et al. (2007) we assume that the intensity M of the underlying Poisson process of grains increases together with the scaling parameter λ as M=λγ, for some γ>0. The results are applicable to the Boolean model and exponential G and rely on an expansion of G in Charlier polynomials and a generalization of Mehler’s formula. Application to solution of Burgers’ equation with initial aggregated random grain data is discussed.

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  • Surgailis, Donatas, 2024. "Scaling limits of nonlinear functions of random grain model, with application to Burgers’ equation," Stochastic Processes and their Applications, Elsevier, vol. 174(C).
    • Handle: RePEc:eee:spapps:v:174:y:2024:i:c:s0304414924000966
    DOI: 10.1016/j.spa.2024.104390
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    References listed on IDEAS

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    1. Surgailis, Donatas, 2020. "Scaling transition and edge effects for negatively dependent linear random fields on Z2," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7518-7546.
    2. Puplinskaitė, Donata & Surgailis, Donatas, 2015. "Scaling transition for long-range dependent Gaussian random fields," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2256-2271.
    3. Hermine Biermé & Anne Estrade & Ingemar Kaj, 2010. "Self-similar Random Fields and Rescaled Random Balls Models," Journal of Theoretical Probability, Springer, vol. 23(4), pages 1110-1141, December.
    4. Lahiri, S.N. & Robinson, Peter M., 2016. "Central limit theorems for long range dependent spatial linear processes," LSE Research Online Documents on Economics 65331, London School of Economics and Political Science, LSE Library.
    5. Pilipauskaitė, Vytautė & Surgailis, Donatas, 2017. "Scaling transition for nonlinear random fields with long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2751-2779.
    6. Pipiras,Vladas & Taqqu,Murad S., 2017. "Long-Range Dependence and Self-Similarity," Cambridge Books, Cambridge University Press, number 9781107039469, September.
    7. Pilipauskaitė, Vytautė & Surgailis, Donatas, 2014. "Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 1011-1035.
    8. Thomas Mikosch & Gennady Samorodnitsky, 2007. "Scaling Limits for Cumulative Input Processes," Mathematics of Operations Research, INFORMS, vol. 32(4), pages 890-918, November.
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