Compatibility results for conditional distributions

In various frameworks, to assess the joint distribution of a k -dimensional random vector X = ( X 1 , … , X k ) , one selects some putative conditional distributions Q 1 , … , Q k . Each Q i is regarded as a possible (or putative) conditional distribution for X i given ( X 1 , … , X i − 1 , X i + 1 , … , X k ) . The Q i are compatible if there is a joint distribution P for X with conditionals Q 1 , … , 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 R . 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 , … , Q k is requested to belong to some given class P 0 of distributions. Two choices for P 0 are considered, that is, P 0 = { exchangeable laws } and P 0 = { laws with identical univariate marginals } .