Open Access
2024 Fluctuations of polyspectra in spherical and Euclidean random wave models
Francesco Grotto, Leonardo Maini, Anna Paola Todino
Author Affiliations +
Electron. Commun. Probab. 29: 1-12 (2024). DOI: 10.1214/24-ECP578


We consider polynomial transforms (polyspectra) of Berry’s model – the Euclidean Random Wave model – and of Random Hyperspherical Harmonics. We determine the asymptotic behavior of variance for polyspectra of any order in the high-frequency limit. In particular, we are able to treat polyspectra of any odd order q5, whose asymptotic behavior was left as a conjecture in the case of Random Hyperspherical Harmonics by Marinucci and Wigman (Comm. Math. Phys. 2014). To this end, we exploit a relation between the variance of polyspectra and the distribution of uniform random walks on Euclidean space with finitely many steps, which allows us to rely on technical results in the latter context.

Funding Statement

F. G. and A. P. T. acknowledge support of INdAM through the INdAM-GNAMPA Projects CUP_E55F22000270001 and CUP E53C23001670001. L.M. acknowledges support of the Luxembourg National Research Fund PRIDE17/1224660/GPS.


F. G. wishes to thank the University of Luxembourg for the warm hospitality during part of the preparation of the present work.


Download Citation

Francesco Grotto. Leonardo Maini. Anna Paola Todino. "Fluctuations of polyspectra in spherical and Euclidean random wave models." Electron. Commun. Probab. 29 1 - 12, 2024.


Received: 16 March 2023; Accepted: 16 February 2024; Published: 2024
First available in Project Euclid: 4 March 2024

Digital Object Identifier: 10.1214/24-ECP578

Primary: 33C10 , 42C10 , 60D05 , 60G50

Keywords: Bessel functions , Gaussian Eigenfunctions , Random walk

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