Abstract
A processX on the setÑ of all finite subsetsJ ofN is said to be spreadable, if\(\left( {X_{pJ} } \right)\mathop = \limits^d \left( {X_J } \right)\) for all subsequencesp=(p 1,p 2,...) ofN, wherepJ={p j ;j∈J}. Spreadable processes are characterized in this paper by a representation formula, similar to those obtained by Aldous and Hoover for exchangeable arrays of r.v.'s. Our representation is equivalent to the statement that a process onÑ is spreadable, iff it can be extended to an exchangeable process indexed by all finite sequences of distinct elements fromN. The latter result may be regarded as a multivariate extension of a theorem by Ryll-Nardzewski, stating that, for infinite sequences of r.v.'s, the notions of exchangeability and spreadability are equivalent.
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Kallenberg, O. Symmetries on random arrays and set-indexed processes. J Theor Probab 5, 727–765 (1992). https://doi.org/10.1007/BF01058727
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DOI: https://doi.org/10.1007/BF01058727