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.
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Kallenberg, O. Exchangeable random measures in the plane. J Theor Probab 3, 81–136 (1990). https://doi.org/10.1007/BF01063330
- Separate and joint exchangeability
- ergodic distributions
- Poisson processes
- uniform random variables