Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Exchangeable random measures in the plane


A random measure ξ on [0,1]2, [0, 1]}ℝ+ or ℝ + 2 is said to be separately exchangeable, if its distribution is invariant under arbitrary Lebesgue measure-preserving transformations in the two coordinates, and jointly exchangeable if ξ is defined on [0,1]2 or ℝ + 2 , and its distribution is invariant under mappings by a common measure-preserving transformation in both directions. In each case, we derive a general representation of ξ in terms of independent Poisson processes and i.i.d. random variables.

This is a preview of subscription content, log in to check access.


  1. 1.

    Aldous, D. J. (1981). Representations for partially exchangeable arrays of random variables.J. Multivar. Anal. 11, 581–598.

  2. 2.

    Aldous, D. J. (1985). Exchangeability and related topics. In Hennequin, P. L. (Ed.),École d'Été de Probabilités de Saint-Flour XIII - 1983, Lectures Notes in Mathematics, Vol. 1117, Springer-Verlag, Berlin, pp. 1–198.

  3. 3.

    Dynkin, E. B. (1978). Sufficient statistics and extreme points.Ann. Prob. 6, 705–730.

  4. 4.

    Hestir, K. (1987). The Aldous representation theorem and weakly exchangeable nonnegative definite arrays. Thesis, Statistics Department, University of California, Berkeley.

  5. 5.

    Hoover, D. N. (1979). Relations on probability spaces and arrays of random variables. Preprint, Institute for Advanced Study, Princeton.

  6. 6.

    Hoover, D. N. (1982). Row-column exchangeability and a generalized model for probability. In Koch, G., and Spizzichino, F. (Eds.),Exchangeability in Probability and Statistics, North-Holland, Amsterdam, pp. 281–291.

  7. 7.

    Kallenberg, O. (1975). On symmetrically distributed random measures.Trans. Am. Math. Soc. 202, 105–121.

  8. 8.

    Kallenberg, O. (1982). The stationarity-invariance problem. In Koch, G., and Spizzichino, F. (Eds.),Exchangeability in Probability and Statistics, North-Holland, Amsterdam, pp. 293–296.

  9. 9.

    Kallenberg, O. (1986).Random Measures, 4th ed. Akademie-Verlag and Academic Press, Berlin-London.

  10. 10.

    Kallenberg, O. (1988). Spreading and predictable sampling in exchangeable sequences and processes.Ann. Prob. 16, 508–534.

  11. 11.

    Kallenberg, O. (1988). Some new representations in bivariate exchangeability.Prob. Theory Rel. Fields 77, 415–455.

  12. 12.

    Kallenberg, O. (1989). On the representation theorem for exchangeable arrays.J. Multivar. Anal. 30, 137–154.

  13. 13.

    Kallenberg, O., and Szulga, J. (1989). Multiple integration with respect to Poisson and Lévy processes.Prob. Theory Rel. Fields, to appear.

  14. 14.

    Krickeberg, K. (1974). Moments of point processes. In Harding, E. F., and Kendall, D. G. (Eds.),Stochatic Geometry, Wiley, New York, pp. 89–113.

  15. 15.

    Maitra, A. (1977). Integral representations of invariant measures.Trans. Am. Math. Soc. 229, 209–225.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kallenberg, O. Exchangeable random measures in the plane. J Theor Probab 3, 81–136 (1990).

Download citation

Key words

  • Separate and joint exchangeability
  • ergodic distributions
  • Poisson processes
  • uniform random variables