Abstract
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.
Highlights
The De Finetti theorem states that any infinite sequence of exchangeable random variables is conditionally i.i.d
We are interested in random selection procedures of elements of E that will be characterized by a family of random variables
If E = IN and the weight function were a constant m, {N (x)}x∈IN would be a sequence of exchangeable random variables and De Finetti’s theorem would ensure that, necessarily, the distribution of {N (x)}x∈E corresponds to the choice of a value θ, with any distribution F on [0, 1], while, conditional on θ, the N (x) are independent with common distribution: P{N (x) = 1 | θ } = θ = ξm/(1−ξ+ξm)
Summary
The De Finetti theorem states that any infinite sequence of exchangeable random variables is conditionally i.i.d. In terms of the {0, 1}-valued stochastic process {N (x)}x∈E, this means being able to specify a set of compatible finite-dimensional distributions satisfying the Assumptions (a) and (b). The first one assumes the existence of a sequence of elements of E with the same weight (the corresponding N (x) are exchangeable) and the conditional independence property is extended to the remaining variables that are no longer identically distributed. The second De Finetti-type result assumes the existence of a sequence of elements in E with weights in IN , and proves the conditional independence of the whole family of random variables {N (x)}x∈E, which are not identically distributed.
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