  Let $X$ be a $p$-variate random vector and $\widetilde{X}$ a knockoff copy of
$X$ (in the sense of \cite{CFJL18}). A new approach for constructing
$\widetilde{X}$ (henceforth, NA) has been introduced in \cite{JSPI}. NA has
essentially three advantages: (i) To build $\widetilde{X}$ is straightforward;
(ii) The joint distribution of $(X,\widetilde{X})$ can be written in closed
form; (iii) $\widetilde{X}$ is often optimal under various criteria. However,
for NA to apply, $X_1,\ldots, X_p$ should be conditionally independent given
some random element $Z$. Our first result is that any probability measure $\mu$
on $\mathbb{R}^p$ can be approximated by a probability measure $\mu_0$ of the
form $$\mu_0\bigl(A_1\times\ldots\times A_p\bigr)=E\Bigl\{\prod_{i=1}^p
P(X_i\in A_i\mid Z)\Bigr\}.$$ The approximation is in total variation distance
when $\mu$ is absolutely continuous, and an explicit formula for $\mu_0$ is
provided. If $X\sim\mu_0$, then $X_1,\ldots,X_p$ are conditionally independent.
Hence, with a negligible error, one can assume $X\sim\mu_0$ and build
$\widetilde{X}$ through NA. Our second result is a characterization of the
knockoffs $\widetilde{X}$ obtained via NA. It is shown that $\widetilde{X}$ is
of this type if and only if the pair $(X,\widetilde{X})$ 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 $\widetilde{X}$ given $X$ is obtained in a few
cases. In one of such cases, it is assumed $X_i\in\{0,1\}$ for all $i$.
