We consider random selection processes of weighted elements in an arbitrary set. Their conditional distributions are shown to be a generalization of the hypergeometric distribution, while the marginal distributions can always be chosen as generalized binomial distributions. Then we propose sufficient conditions on the weight function ensuring that the marginal distributions are necessarily of the generalized binomial form. In these cases, the corresponding indicator random variables are conditionally independent (as in the classical De Finetti theorem) though they are neither exchangeable nor identically distributed.

We introduce the concept of a random assignment process (roughly speaking, such a process assigns, according to some weight function, labels to the points of an arbitrary set), and we impose conditions on the weight function ensuring that a De Finetti-type theorem is satisfied. In particular, this provides a wide class of finite-valued random variables which, despite they are neither exchangeable nor identically distributed, verify a De Finetti-type theorem (i.e., they are conditionally independent).

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