Consider $d$ random variables. For each set of $k$ variables, we are given a joint probability distribution. We want to know that whether these distributions correspond to a valid joint probability distribution of all $d$ variables. We can assume that each variable has a finite domain.


I think a necessary condition is that, all given distributions should agree with the same lower dimensional distributions when we integrates some variables out. But this seems not a sufficient condition.


Is there any simple necessary and sufficient condition? or can we find a simple but stronger necessary condition? or is the above necessary condition in fact sufficient? Thanks.



There are d random variables. Given all k-D joint probability distributions with some k<d, what is the necessary and sufficient condition for these distributions to be feasible?
I asked myself the very same question some time ago. First, let me show that the obvious necessary condition is not sufficient.


Let $X\_1,Y\_1,Y\_2,Z\_2,Z\_3,X\_3$, be six random variables having the same non-deterministic law such that: $X\_1=Y\_1$, $Y\_2=Z\_2$ and $(Z\_3,X\_3)$ are independent. Then there cannot exist $(X,Y,Z)$ such that $(X,Y)\sim (X\_1,Y\_1)$, $(Y,Z)\sim (Y\_2,Z\_2)$ and $(Z,X)\sim (Z\_3,X\_3)$.


Now, there is a condition that, added to the sub-dimensional joint law correspondence is sufficient for feasability. It is written in a short note 
<http://www-fourier.ujf-grenoble.fr/~bkloeckn/papiers/compatibility.pdf> on my web page in the case of three variables; the full condition is stated and proved by Hans G. Kellerer.
in *Verteilungsfunktionen mit gegebenen Marginalverteilungen*. (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 3:247–270 (1964), 1964.). Note if you do not recognize the name of the journal that its the former name of PTRF (Probability Theory and Related Fields). I do not know a reference in english, but you can have a look at the MR review of that paper.


