On the representation theorem for exchangeable arrays

Aldous and Hoover have proved independently that an array X = (Xij, i, j [set membership, variant] ) of random variables is exchangeable under separate or joint permutations of rows and columns, iff a.s. Xij[reverse not equivalent]f([alpha], [xi]i, [eta]j, [xi]ij) or Xij[reverse not equivalent]f([alpha], [xi]i, [xi]j, [xi]ij), respectively, for some measurable function f: 4--> and some i.i.d. random variables [alpha], [xi]i, [eta]j, [xi]ij, i, j[set membership, variant], or [alpha], [xi]i, [xi]ij=[xi]ji, 1<=i<=j. Hoover also gave a criterion for two functions f and g as above to give rise to arrays with the same distribution. The aim of this paper is to give an elementary proof of Hoover's result, and to deduce some further equivalence criteria. The present methods will also provide a simple approach to certain conditional properties of exchangeable arrays.