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

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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.

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Notes

  1. For more details on this, and more, we refer the reader to [2].

  2. For any \(\lambda >0\), we have \(\mathcal{L}_k(\lambda M) = \lambda ^k \mathcal{L}_k(M)\), where \(\lambda M = \{\lambda x: x\in M\}\).

  3. We refer to [4] for further discussion on connections between analytical assumptions and regularity conditions for Gaussian random fields.

  4. By a symmetric set, we mean that every chord passing through the origin must be bisected at the origin.

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Acknowledgements

Both authors kindly acknowledge the financial support received from IFCAM (Indo-French Center for Applied Mathematics) to work on this project in India (TIFR-CAM, Bangalore) and in France (ESSEC Business school, Paris) in 2014 and 2015. This result has been presented at EVA conference (invited ‘RARE’ session) in June 2015. This study has also received the support from the European Union’s Seventh Framework Programme for research, technological development and demonstration under Grant Agreement No 318984-RARE, and from the Airbus Foundation Chair on Mathematics of Complex Systems at TIFR-CAM, Bangalore. Note that another study on the same topic [20] has been worked out in parallel providing the same result.

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Authors and Affiliations

  1. ESSEC Business School, CREAR, Av. Bernard Hirsch, 95021, Cergy-Pontoise Cedex, France

    Marie Kratz

  2. TIFR–CAM, Bangalore, India

    Sreekar Vadlamani

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  1. Marie Kratz
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  2. Sreekar Vadlamani
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Correspondence to Sreekar Vadlamani.

Additional information

This work was supported by the IFCAM project titled “Geometry of excursion sets of random fields”.

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Kratz, M., Vadlamani, S. Central Limit Theorem for Lipschitz–Killing Curvatures of Excursion Sets of Gaussian Random Fields. J Theor Probab 31, 1729–1758 (2018). https://doi.org/10.1007/s10959-017-0760-6

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  • DOI: https://doi.org/10.1007/s10959-017-0760-6

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