Fourier coefficients of invariant random fields on homogeneous spaces of compact Lie groups

Baldi, P.; Trapani, S.

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 648-671.

Soit $T$ un champ aléatoire invariant par rapport à l’action d’un groupe compact $G$. On étudie les propriétés de ses coefficients de Fourier telles que l’orthogonalité et la gaussianité. En particulier on établit des conditions qui garantissent que l’indépendance de ces coefficients entraîne qu’ils sont gaussiens. Une conséquence remarquable est que, en général, il n’est pas possible de générer par simulation un champ aléatoire non gaussien invariant à l’aide de son développement par des coefficients indépendants.

Let $T$ be a random field invariant under the action of a compact group $G$. In the line of previous work we investigate properties of the Fourier coefficients as orthogonality and Gaussianity. In particular we give conditions ensuring that independence of the random Fourier coefficients implies Gaussianity. As a consequence, in general, it is not possible to simulate a non-Gaussian invariant random field through its Fourier expansion using independent coefficients.

Classification : 60B15, 60E05, 43A30
Mots clés : invariant random fields, Fourier expansions, characterization of gaussian random fields

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