Compatibility results for conditional distributions

In various frameworks, to assess the joint distribution of a $k$-dimensional random vector $X=(X_1,ldots,X_k)$, one selects some putative conditional distributions $Q_1,ldots,Q_k$. Each $Q_i$ is regarded as a possible (or putative) conditional distribution for $X_i$ given $(X_1,ldots,X_i-1,X_i+1,ldots,X_k)$. The $Q_i$ are compatible if there is a joint distribution $P$ for $X$ with conditionals $Q_1,ldots,Q_k$. Three types of compatibility results are given in this paper. First, the $X_i$ are assumed to take values in compact subsets of $mathbbR$. Second, the $Q_i$ are supposed to have densities with respect to reference measures. Third, a stronger form of compatibility is investigated. The law $P$ with conditionals $Q_1,ldots,Q_k$ is requested to belong to some given class $mathcalP_0$ of distributions. Two choices for $mathcalP_0$ are considered, that is, $mathcalP_0=$exchangeable laws$$ and $mathcalP_0=$laws with identical univariate marginals$$.