Spherical symmetry: An elementary justification

  • Morris L. Eaton
  • Sandra Fortini
  • Eugenio Regazzini


The present paper includes characterizations of the conditions of spherical symmetry and of centered spherical symmetry. These characterizations provide an empirical justification for the above mentioned conditions of symmetry.


exchangeability [centered] spherical symmetry mixture of normal laws 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Billingsely, P. (1986).Probability and Measure. Wiley, New York.Google Scholar
  2. Breiman, L. (1968).Probability. Addison-Wesley, Reading, Massachussetts.zbMATHGoogle Scholar
  3. Bühlmann, H. (1958). Le problème “limite centrale” pour les variables aléatoire échangeables.C. R. Acad. Sci. Paris 246, 534–536.zbMATHMathSciNetGoogle Scholar
  4. Bühlmann, H. (1960). Austauschbare stochastische variabeln und ihre grenzwertsätze.Univ. Calif. Publ. Statist. 3, 1–35.Google Scholar
  5. Diaconis, P., Eaton, M. L., Lauritzen, S. L. (1992). Finite de Finetti theorems in linear models and multivariata analysis.Scand. Jour. Statist. 19, 289–315.zbMATHMathSciNetGoogle Scholar
  6. Eaton, M. L. (1989).Group Invariance Applications in Statistics. Regional Conference Series in Prob. and Statist., Vol. 1. IMS, ASA.Google Scholar
  7. de Finetti, B. (1934), Come giustificare elementarmente la «legge normale» della probabilità?Periodico delle Matematiche Serie IV 14, 197–210. Reprint inArchimede 42 (1990), 51–64.zbMATHGoogle Scholar
  8. de Finetti, B. (1938), Sur la condition d'equivalence partielle.Act. Scient. Ind. 739, 5–18.Google Scholar
  9. Kingman, J. F. C. (1972). On random sequences with spherical symmetry.Biometrika 59, 492–494.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Link, G. (1980). Representation theorems of de Finetti type for (partially) symmetric probability measures. InStudies in Inductive Logic and Probability, II (C. Jeffrey, ed.) University of California Press, Berkley.Google Scholar
  11. Schoenberg, I. J. (1938). Metric spaces and positive definite functions.Trans. Amer. Math. Soc. 44, 522–536.zbMATHCrossRefMathSciNetGoogle Scholar
  12. Smith, A. F. M. (1981). On random sequences with centered spherical symmetry.J. R. Statist. Soc. B 43, 208–209.zbMATHGoogle Scholar

Copyright information

© Società Italiana di Statistica 1993

Authors and Affiliations

  • Morris L. Eaton
    • 1
  • Sandra Fortini
    • 2
  • Eugenio Regazzini
    • 3
  1. 1.School of StatisticsUniversity of MinnesotaMinneapolisUSA
  2. 2.CNR-IAMIMilanoItaly
  3. 3.Università «L. Bocconi»MilanoItaly

Personalised recommendations