Summary
According to a theorem of de Finetti's, an exchangeable stochastic process with values in a compact metric space can be represented as a mixture of sequences of independent, identically distributed random variables. This paper demonstrates the existence of a separable metric space for which the conclusion fails. In the opposite direction, an example is given of a nonstandard space for which the representation necessarily holds.
Modifications of the argument lead to examples of exchangeable stochastic processes and stationary Markov processes which take values in a separable metric space but do not satisfy the conclusions of the Kolmogorov consistency theorem.
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Research partially supported by National Science Foundation Grant MCS 77-01665
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Dubins, L.E., Freedman, D.A. Exchangeable processes need not be mixtures of independent, identically distributed random variables. Z. Wahrscheinlichkeitstheorie verw Gebiete 48, 115–132 (1979). https://doi.org/10.1007/BF01886868
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DOI: https://doi.org/10.1007/BF01886868