Compatibility results for conditional distributions

In various frameworks, to assess the joint distribution of a k-dimensional random vector X = (X1, . . . ,Xk), one selects some putative conditional distributions Q1, . . . ,Qk. Each Qi is regarded as a possible (or putative) conditional distribution for Xi given (X1, . . . ,Xi−1,Xi+1, . . . ,Xk). The Qi are compatible if there is a joint distribution P for X with conditionals Q1, . . . ,Qk. Three types of compatibility results are given in this paper. First, the Xi are assumed to take values in compact subsets of R. Second, the Qi are supposed to have densities with respect to reference measures. Third, a stronger form of compatibility is investigated. The law P with conditionals Q1, . . . ,Qk is requested to belong to some given class P0 of distributions. Two choices for P0 are considered, that is, P0 = {exchangeable laws} and P0 = {laws with identical univariate marginals}.