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Central Limit Theorem for Lipschitz–Killing Curvatures of Excursion Sets of Gaussian Random Fields

Author

Listed:
  • Marie Kratz

    (CREAR)

  • Sreekar Vadlamani

    (TIFR–CAM)

Abstract

Our interest in this paper is to explore limit theorems for various geometric functionals of excursion sets of isotropic Gaussian random fields. In the past, asymptotics of nonlinear functionals of Gaussian random fields have been studied [see Berman (Sojourns and extremes of stochastic processes, Wadsworth & Brooks, Monterey, 1991), Kratz and León (Extremes 3(1):57–86, 2000), Kratz and León (J Theor Probab 14(3):639–672, 2001), Meshenmoser and Shashkin (Stat Probab Lett 81(6):642–646, 2011), Pham (Stoch Proc Appl 123(6):2158–2174, 2013), Spodarev (Chapter in modern stochastics and applications, volume 90 of the series Springer optimization and its applications, pp 221–241, 2013) for a sample of works in such settings], the most recent addition being (Adler and Naitzat in Stoch Proc Appl 2016; Estrade and León in Ann Probab 2016) where a central limit theorem (CLT) for Euler integral and Euler–Poincaré characteristic, respectively, of the excursions set of a Gaussian random field is proven under some conditions. In this paper, we obtain a CLT for some global geometric functionals, called the Lipschitz–Killing curvatures of excursion sets of Gaussian random fields, in an appropriate setting.

Suggested Citation

  • Marie Kratz & Sreekar Vadlamani, 2018. "Central Limit Theorem for Lipschitz–Killing Curvatures of Excursion Sets of Gaussian Random Fields," Journal of Theoretical Probability, Springer, vol. 31(3), pages 1729-1758, September.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:3:d:10.1007_s10959-017-0760-6
    DOI: 10.1007/s10959-017-0760-6
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    References listed on IDEAS

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    1. Nourdin, Ivan & Peccati, Giovanni & Podolskij, Mark, 2011. "Quantitative Breuer-Major theorems," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 793-812, April.
    2. Kratz, Marie F. & León, JoséR., 1997. "Hermite polynomial expansion for non-smooth functionals of stationary Gaussian processes: Crossings and extremes," Stochastic Processes and their Applications, Elsevier, vol. 66(2), pages 237-252, March.
    3. Pham, Viet-Hung, 2013. "On the rate of convergence for central limit theorems of sojourn times of Gaussian fields," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2158-2174.
    4. Meschenmoser, D. & Shashkin, A., 2011. "Functional central limit theorem for the volume of excursion sets generated by associated random fields," Statistics & Probability Letters, Elsevier, vol. 81(6), pages 642-646, June.
    5. 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.
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