Reproducibility of multivariate probability distributions and related properties

In this work, we consider two related concepts: conditional quantile reproducibility of multivariate probability distributions and reproducibility of the multvariate distributions themselves. We put forward several results demonstrating how one of the reproducibility types can lead to another. We then give a theorem that suggests a method of going up from a fully reproducible n-dimensional distribution to a distribution of (n+1) dimensions with a fully reproducible conditional quantile.