De Finetti-type theorems for random selection processes. Necessary and sufficient conditions

This paper is concerned with random selection processes for which, in the previous literature, several sufficient conditions for a De Finetti-type theorem (granting the conditional independence of a family of random variables) have been proposed. In this paper, we prove that these conditions are also necessary. In this sense, we identify the extremes of the convex set of marginal distributions of a random selection process, and then we give a thorough description and classification of such processes.