Electronic Communications in Probability

Mean-Square continuity on homogeneous spaces of compact groups

Domenico Marinucci and Giovanni Peccati

Full-text: Open access


We show that any finite-variance, isotropic random field on a compact group is necessarily mean-square continuous, under standard measurability assumptions. The result extends to isotropic random fields defined on homogeneous spaces where the group acts continuously.

Article information

Electron. Commun. Probab. Volume 18 (2013), paper no. 37, 10 pp.

Accepted: 23 May 2013
First available in Project Euclid: 7 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G05: Foundations of stochastic processes
Secondary: 60G60: Random fields

Random Processes Isotropy Mean-Square Continuity

This work is licensed under a Creative Commons Attribution 3.0 License.


Marinucci, Domenico; Peccati, Giovanni. Mean-Square continuity on homogeneous spaces of compact groups. Electron. Commun. Probab. 18 (2013), paper no. 37, 10 pp. doi:10.1214/ECP.v18-2400. http://projecteuclid.org/euclid.ecp/1465315576.

Export citation


  • Adler, Robert J. The geometry of random fields. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Ltd., Chichester, 1981. xi+280 pp. ISBN: 0-471-27844-0
  • Adler, Robert J.; Taylor, Jonathan E. Random fields and geometry. Springer Monographs in Mathematics. Springer, New York, 2007. xviii+448 pp. ISBN: 978-0-387-48112-8
  • Baldi, Paolo; Marinucci, Domenico. Some characterizations of the spherical harmonics coefficients for isotropic random fields. Statist. Probab. Lett. 77 (2007), no. 5, 490–496.
  • Baldi, P.; Marinucci, D.; Varadarajan, V. S. On the characterization of isotropic Gaussian fields on homogeneous spaces of compact groups. Electron. Comm. Probab. 12 (2007), 291–302.
  • Crum, M. M. On positive-definite functions. Proc. London Math. Soc. (3) 6 (1956), 548–560.
  • S. Dodelson (2003). Modern Cosmology. Academic Press.
  • Dudley, R. M. Real analysis and probability. Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002. x+555 pp. ISBN: 0-521-00754-2
  • Duistermaat, J. J.; Kolk, J. A. C. Lie groups. Universitext. Springer-Verlag, Berlin, 2000. viii+344 pp. ISBN: 3-540-15293-8
  • R. Durrer (2008). The Cosmic Microwave Background. Cambridge University Press.
  • Gneiting, Tilmann; Sasvári, Zoltán. The characterization problem for isotropic covariance functions. Math. Geol. 31 (1999), no. 1, 105–111.
  • James, Gordon; Liebeck, Martin. Representations and characters of groups. Second edition. Cambridge University Press, New York, 2001. viii+458 pp. ISBN: 0-521-00392-X
  • Kallianpur, Gopinath. Stochastic filtering theory. Applications of Mathematics, 13. Springer-Verlag, New York-Berlin, 1980. xvi+316 pp. ISBN: 0-387-90445-X
  • Leonenko, Nikolai. Limit theorems for random fields with singular spectrum. Mathematics and its Applications, 465. Kluwer Academic Publishers, Dordrecht, 1999. viii+401 pp. ISBN: 0-7923-5635-7
  • Marinucci, Domenico; Peccati, Giovanni. Random fields on the sphere. Representation, limit theorems and cosmological applications. London Mathematical Society Lecture Note Series, 389. Cambridge University Press, Cambridge, 2011. xii+341 pp. ISBN: 978-0-521-17561-6
  • A. M. Obukhov (1947). Statistically homogeneous random fields on a sphere. Uspehi Mat. Nauk 2, no. 2, 196–198.
  • Peccati, G.; Pycke, J.-R. Decompositions of stochastic processes based on irreducible group representations. Teor. Veroyatn. Primen. 54 (2009), no. 2, 304–336; translation in Theory Probab. Appl. 54 (2010), no. 2, 217–245
  • Sasvári, Zoltán. Positive definite and definitizable functions. Mathematical Topics, 2. Akademie Verlag, Berlin, 1994. 208 pp. ISBN: 3-05-501446-4
  • Schoenberg, I. J. Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44 (1938), no. 3, 522–536.
  • H.J. Starkloff (2009). Notes on Stationary Processes and Fields, Oral Presentation, available online at http://whz-cms-10.zw.fh-zwickau.de/hast/Texte/starkloff_frbg20090915.pdf
  • Varadarajan, V. S. An introduction to harmonic analysis on semisimple Lie groups. Corrected reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 16. Cambridge University Press, Cambridge, 1999. x+316 pp. ISBN: 0-521-34156-6; 0-521-66362-8
  • Varshalovich, D. A.; Moskalev, A. N.; KhersonskiÄ­, V. K. Quantum theory of angular momentum. Irreducible tensors, spherical harmonics, vector coupling coefficients, $3nj$ symbols. Translated from the Russian. World Scientific Publishing Co., Inc., Teaneck, NJ, 1988. xii+514 pp. ISBN: 9971-50-107-4
  • Vilenkin, N. Ja.; Klimyk, A. U. Representation of Lie groups and special functions. Vol. 1. Simplest Lie groups, special functions and integral transforms. Translated from the Russian by V. A. Groza and A. A. Groza. Mathematics and its Applications (Soviet Series), 72. Kluwer Academic Publishers Group, Dordrecht, 1991. xxiv+608 pp. ISBN: 0-7923-1466-2
  • Yadrenko, M. Ĭ. Spectral theory of random fields. Translated from the Russian. Translation Series in Mathematics and Engineering. Optimization Software, Inc., Publications Division, New York, 1983. iii+259 pp. ISBN: 0-911575-00-6
  • Yaglom, A. M. Second-order homogeneous random fields. 1961 Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II pp. 593–622 Univ. California Press, Berkeley, Calif.