We present a detailed proof of the Dovbysh-Sudakov representation for symmetric positive definite weakly exchangeable infinite random arrays, called Gram-de Finetti matrices, which is based on the representation result of Aldous and Hoover for arbitrary (not necessarily positive definite) symmetric weakly exchangeable arrays.
"On the Dovbysh-Sudakov representation result." Electron. Commun. Probab. 15 330 - 338, 2010. https://doi.org/10.1214/ECP.v15-1562