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2021 Fluctuations of the Gromov–Prohorov sample model
Jacques de Catelan, Pierre-Loïc Méliot
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Electron. J. Probab. 26: 1-37 (2021). DOI: 10.1214/21-EJP634


In this paper, we study the fluctuations of observables of metric measure spaces which are random discrete approximations Xn of a fixed arbitrary (complete, separable) metric measure space X=(X,d,μ). These observables Φ(Xn) are polynomials in the sense of Greven–Pfaffelhuber–Winter, and we show that for a generic model space X, they yield asymptotically normal random variables. However, if X is a compact homogeneous space, then the fluctuations of the observables are much smaller, and after an adequate rescaling, they converge towards probability distributions which are not Gaussian. Conversely, we prove that if all the fluctuations of the observables Φ(Xn) are smaller than in the generic case, then the measure metric space X is compact homogeneous. The proofs of these results rely on the Gromov reconstruction principle, and on an adaptation of the method of cumulants and mod-Gaussian convergence developed by Féray–Méliot–Nikeghbali. As an application of our results, we construct a statistical test of the hypothesis of symmetry of a compact Riemannian manifold.


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Jacques de Catelan. Pierre-Loïc Méliot. "Fluctuations of the Gromov–Prohorov sample model." Electron. J. Probab. 26 1 - 37, 2021.


Received: 20 May 2020; Accepted: 27 April 2021; Published: 2021
First available in Project Euclid: 6 May 2021

Digital Object Identifier: 10.1214/21-EJP634

Primary: 60B05 , 60B10 , 60F05

Keywords: combinatorics of the cumulants of random variables , discrete approximation of metric spaces , Gromov–Prohorov topology


Vol.26 • 2021
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