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Asymptotic behaviour of level sets of needlet random fields

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  • Shevchenko, Radomyra
  • Todino, Anna Paola

Abstract

We consider sequences of needlet random fields defined as weighted averaged forms of spherical Gaussian eigenfunctions. Our main result is a Central Limit Theorem in the high energy setting, for the boundary lengths of their excursion sets. This result is based on Wiener chaos expansion and Stein–Malliavin techniques for nonlinear functionals of random fields. To this end, a careful analysis of the variances of each chaotic component of the boundary length is carried out, showing that they are asymptotically constant, after normalisation, for all terms of the expansion and no leading component arises.

Suggested Citation

  • Shevchenko, Radomyra & Todino, Anna Paola, 2023. "Asymptotic behaviour of level sets of needlet random fields," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 268-318.
    • Handle: RePEc:eee:spapps:v:155:y:2023:i:c:p:268-318
    DOI: 10.1016/j.spa.2022.10.011
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    References listed on IDEAS

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    1. Marie F. Kratz & José R. León, 2001. "Central Limit Theorems for Level Functionals of Stationary Gaussian Processes and Fields," Journal of Theoretical Probability, Springer, vol. 14(3), pages 639-672, July.
    2. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
    3. Lan, Xiaohong & Marinucci, Domenico, 2009. "On the dependence structure of wavelet coefficients for spherical random fields," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3749-3766, October.
    4. Marie Kratz & Sreekar Vadlamani, 2016. "CLT for Lipschitz-Killing curvatures of excursion sets of Gaussian random fields," Working Papers hal-01373091, HAL.
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