How to Construct a Two Dimensional Random Vector with a Given Conditional Structure

Abstract

A set P ⊂ X × M ₁(Y) (X,Y Polish spaces, M ₁(Y) the set of Borei probability measures over Y) is considered. A random vector (ξ, η) with values in X × Y is called a P-vector, a rich P-vector and an extremal P-vector if (almost surely w.r.t $\mathcal {L} (\xi )) \mathcal {L}(\eta |\xi = x) \in \mathcal{P}_x, \mathcal{L} (\eta |\xi = x)$ has the maximal possible support among the distributions in $\mathcal{P}_x \textup{and} \mathcal {L} (\xi )) \mathcal {L}(\eta |\xi = x) \in \mathcal{P}_x, \mathcal{L} (\eta |\xi = x)$ is an extremal measure in $\mathcal{P}_x$, respectively. The purpose of the paper is to find sufficient conditions for the existence of P-vectors, to cover the situations when the sections $\mathcal{P}_x$ are denned by momenta marginal problems¹.