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November 2018 A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions
Valentina Cammarota, Domenico Marinucci
Ann. Probab. 46(6): 3188-3228 (November 2018). DOI: 10.1214/17-AOP1245

Abstract

We establish here a quantitative central limit theorem (in Wasserstein distance) for the Euler–Poincaré characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler–Poincaré characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level $u$ is fully degenerate, that is, the Euler–Poincaré characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. We discuss also a possible unifying framework for the Lipschitz–Killing curvatures of the excursion sets for Gaussian spherical harmonics.

Citation

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Valentina Cammarota. Domenico Marinucci. "A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions." Ann. Probab. 46 (6) 3188 - 3228, November 2018. https://doi.org/10.1214/17-AOP1245

Information

Received: 1 April 2016; Revised: 1 June 2017; Published: November 2018
First available in Project Euclid: 25 September 2018

zbMATH: 06975485
MathSciNet: MR3857854
Digital Object Identifier: 10.1214/17-AOP1245

Subjects:
Primary: 33C55, 42C10, 53C65, 60G60, 62M15

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 6 • November 2018
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