An extension of the factorization theorem to the non-positive case

This paper presents a method of determining joint distributions by known conditional distributions. A generalization of the Factorization Theorem is proposed. The generalized theorem is proved under the assumption that the support of unknown joint distribution may be divided into a countable number of sets, which all satisfy the relative weak positivity condition. This condition is defined in the paper and it generalizes the positivity condition introduced by Hammersley and Clifford. The theorem is illustrated with three examples. In the first example we determine a joint density in the case when the support of an unknown density is a continuous nonproduct set from Euclidean space R2. In the second example we seek the joint probability for the number of trials and the number of successes in Bernoulliʹs scheme. We also examine a simple example given by Kaiser and Cressie (J. Multivariate Anal. 73 (2000) 199).