Generating knockoffs via conditional independence

Let X be a p-variate random vector and X a knockoff copy of X (in the sense of [9]). A new approach for constructing X (henceforth, NA) has been introduced in [8]. NA has essentially three advantages: (i) To build X is straightforward; (ii) The joint distribution of (X, X) can be written in closed form; (iii) X is often optimal under various criteria. However, for NA to apply, X1, ... , Xp should be conditionally independent given some random element Z. Our first result is that any probability measure mu on Rp can be approximated by a probability measure mu 0 of the form ?p ? pi mu 0 (A1 x center dot center dot center dot x Ap) = E P(XiE Ai | Z) . i=1 The approximation is in total variation distance when mu is absolutely continuous, and an explicit formula for mu 0 is provided. If X similar to mu 0, then X1, ... , Xp are conditionally independent. Hence, with a negligible error, one can assume X similar to mu 0 and build X ? through NA. Our second result is a characterization of the knockoffs X obtained via NA. It is shown that X is of this type if and only if the pair (X, X) can be extended to an infinite sequence so as to satisfy certain invariance conditions. The basic tool for proving this fact is de Finetti's theorem for partially exchangeable sequences. In addition to the quoted results, an explicit formula for the conditional distribution of X ? given X is obtained in a few cases. In one of such cases, it is assumed Xi E {0, 1} for all i.