Soit un champ aléatoire invariant par rapport à l’action d’un groupe compact . 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 be a random field invariant under the action of a compact group . 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.
Mots clés : invariant random fields, Fourier expansions, characterization of gaussian random fields
@article{AIHPB_2015__51_2_648_0, author = {Baldi, P. 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}, pages = {648--671}, publisher = {Gauthier-Villars}, volume = {51}, number = {2}, year = {2015}, doi = {10.1214/14-AIHP600}, mrnumber = {3335020}, zbl = {1353.60008}, language = {en}, url = {http://www.numdam.org/articles/10.1214/14-AIHP600/} }
TY - JOUR AU - Baldi, P. AU - Trapani, S. TI - Fourier coefficients of invariant random fields on homogeneous spaces of compact Lie groups JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 648 EP - 671 VL - 51 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/14-AIHP600/ DO - 10.1214/14-AIHP600 LA - en ID - AIHPB_2015__51_2_648_0 ER -
%0 Journal Article %A Baldi, P. %A Trapani, S. %T Fourier coefficients of invariant random fields on homogeneous spaces of compact Lie groups %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 648-671 %V 51 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/14-AIHP600/ %R 10.1214/14-AIHP600 %G en %F 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, Tome 51 (2015) no. 2, pp. 648-671. doi : 10.1214/14-AIHP600. http://www.numdam.org/articles/10.1214/14-AIHP600/
[1] Some characterizations of the spherical harmonics coefficients for isotropic random rields. Statist. Probab. Lett. 77 (2007) 490–496. | DOI | MR | Zbl
and .[2] On the characterization of isotropic Gaussian fields on homogeneous spaces of compact groups. Electron. Commun. Probab. 12 (2007) 291–302 (electronic). | DOI | MR | Zbl
, and .[3] T. Bröcker and T. tom Dieck. Representations of Compact Lie Groups. Graduate Texts in Mathematics 98. Springer, New York, 1995. | MR | Zbl
[4] Exercises in Probability. Cambridge Series in Statistical and Probabilistic Mathematics 13. Cambridge Univ. Press, Cambridge, 2003. | DOI | MR | Zbl
and .[5] Analysis on Lie Groups. Cambridge Studies in Advanced Mathematics 110. Cambridge Univ. Press, Cambridge, 2008. | DOI | MR | Zbl
.[6] A characterization of the multivariate normal distribution. Ann. Math. Statist. 33 (1962) 533–541. | DOI | MR | Zbl
and .[7] Characterization Problems in Mathematical Statistics. Wiley, New York, 1973. | MR | Zbl
, and .[8] Invariant random fields in vector bundles and application to cosmology. Ann. Inst. Henri Poincare Probab. Stat. 47 (4) (2011) 1068–1095. | Numdam | MR | Zbl
.[9] Random Fields. London Mathematical Society Lecture Note Series 389. Cambridge Univ. Press, Cambridge, 2011. | MR | Zbl
and .[10] Mean-square continuity on homogeneous spaces of compact groups. Electron. Commun. Probab. 18 (2013) 37. | MR | Zbl
and .[11] Decompositions of stochastic processes based on irreducible group representations. Teor. Veroyatn. Primen. 54 (2) (2009) 304–336. | MR | Zbl
and .[12] Representation of Lie Groups and Special Functions. Vol. 1. Mathematics and Its Applications (Soviet Series) 72. Kluwer Academic, Dordrecht, 1991. | DOI | MR | Zbl
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