Abstract
Our key result is the characterization of exchangeable sequences as being strongly stationary, i.e. invariant in distribution under stopping time shifts. From this we prove homogeneity characterizations of pure and mixed Markov chains. These results carry over to continuous time processes and random sets, and on a whole, our theory provides a unified approach to exchangeability. The key result above is closely related to Dacunha-Castelle's embedding characterization of exchangeability, which is partially extended here to processes on [0, 1]. In the other direction, we prove that a previsible sample from a finite or infinite exchangeable sequence X may be embedded into a copy of X. We finally establish some uniqueness results for exponentially and uniformly killed exchangeable random sets.
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