Fourier coefficients of invariant random fields on homogeneous spaces of compact Lie groups
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, p. 648-671

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.

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.

DOI : https://doi.org/10.1214/14-AIHP600
Classification:  60B15,  60E05,  43A30
Keywords: invariant random fields, Fourier expansions, characterization of gaussian random fields
@article{AIHPB_2015__51_2_648_0,
     author = {Baldi, Paolo and Trapani, S.},
     title = {Fourier coefficients of invariant random fields on homogeneous spaces of compact Lie groups},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {2},
     year = {2015},
     pages = {648-671},
     doi = {10.1214/14-AIHP600},
     mrnumber = {3335020},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_2_648_0}
}
Baldi, P.; Trapani, S. Fourier coefficients of invariant random fields on homogeneous spaces of compact Lie groups. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, pp. 648-671. doi : 10.1214/14-AIHP600. http://www.numdam.org/item/AIHPB_2015__51_2_648_0/

[1] P. Baldi and D. Marinucci. Some characterizations of the spherical harmonics coefficients for isotropic random rields. Statist. Probab. Lett. 77 (2007) 490–496. | MR 2344633 | Zbl 1117.60053

[2] P. Baldi, D. Marinucci and V. S. Varadarajan. On the characterization of isotropic Gaussian fields on homogeneous spaces of compact groups. Electron. Commun. Probab. 12 (2007) 291–302 (electronic). | MR 2342708 | Zbl 1128.60039

[3] T. Bröcker and T. tom Dieck. Representations of Compact Lie Groups. Graduate Texts in Mathematics 98. Springer, New York, 1995. | MR 1410059 | Zbl 0874.22001

[4] L. Chaumont and M. Yor. Exercises in Probability. Cambridge Series in Statistical and Probabilistic Mathematics 13. Cambridge Univ. Press, Cambridge, 2003. | MR 2016344 | Zbl 1065.60001

[5] J. Faraut. Analysis on Lie Groups. Cambridge Studies in Advanced Mathematics 110. Cambridge Univ. Press, Cambridge, 2008. | MR 2426516 | Zbl 1147.22001

[6] S. G. Ghurye and I. Olkin. A characterization of the multivariate normal distribution. Ann. Math. Statist. 33 (1962) 533–541. | MR 137201 | Zbl 0109.13401

[7] A. M. Kagan, Y. V. Linnik and C. Radhakrishna Rao. Characterization Problems in Mathematical Statistics. Wiley, New York, 1973. | MR 346969 | Zbl 0271.62002

[8] A. Malyarenko. Invariant random fields in vector bundles and application to cosmology. Ann. Inst. Henri Poincare Probab. Stat. 47 (4) (2011) 1068–1095. | Numdam | MR 2884225 | Zbl 1268.60072

[9] D. Marinucci and G. Peccati. Random Fields. London Mathematical Society Lecture Note Series 389. Cambridge Univ. Press, Cambridge, 2011. | MR 2840154 | Zbl 1260.60004

[10] D. Marinucci and G. Peccati. Mean-square continuity on homogeneous spaces of compact groups. Electron. Commun. Probab. 18 (2013) 37. | MR 3064996 | Zbl 06346886

[11] G. Peccati and J.-R. Pycke. Decompositions of stochastic processes based on irreducible group representations. Teor. Veroyatn. Primen. 54 (2) (2009) 304–336. | MR 2761557 | Zbl 1229.60039

[12] N. J. Vilenkin and A. U. Klimyk. Representation of Lie Groups and Special Functions. Vol. 1. Mathematics and Its Applications (Soviet Series) 72. Kluwer Academic, Dordrecht, 1991. | MR 1143783 | Zbl 0742.22001