Canonical row-column-exchangeable arrays
It has been shown by \textit{D. J. Aldous} [ibid. 11, 581-598 (1981; Zbl 0474.60044)] that a row-column-exchangeable array \((X_{ij}:\) i,\(j\geq 1)\) of random variables is equivalent in distribution to a standard array of the form \((X^*_{ij})=(f(\alpha,\xi_ i,\eta_ j,\lambda_{ij}))\), where f is a Borel function and \(\alpha,\xi_ i,\eta_ j,\lambda_{ij},i,j=1,2,..\). are independent uniformly distributed random variables. It is shown here that there exists a canonical version \((X'_{ij})=(g(\alpha',\xi '_ i,\eta '_ j,\lambda'_{ij}))\) such that the \(X'_{ij}\)'s and the \(\sigma\)-field \({\mathcal F}'= \sigma(\alpha',\xi'_ i,\eta'_ j: i,j\geq 1)\) are conditionally independent, given the shell \(\sigma\)-field \(\cap_{n>o}\sigma(X'_{ij}: \max(i,j)>n)\). (The result is mis-stated in the text). This statement, though weaker than Aldous's analogous result for the original array is easier to prove and is sufficient for certain characterisation purposes. The proofs rely on Aldous's constructions.
scientific article; zbMATH DE number 3874328