EXCHANGEABLE PROCESSES NEED NOT BE MIXTURES OF INDEPENDENT, IDENTICALLY DISTRIBUTED RANDOM-VARIABLES

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. © 1979 Springer-Verlag.