Symmetries on random arrays and set-indexed processes

A processX on the setÑ of all finite subsetsJ ofN is said to be spreadable, if for all subsequencesp=(p ₁,p ₂,...) ofN, wherepJ={p _j ;jJ}. 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.