Another definition of positive correlation between random variables

Question

I'm reading something about Stein-Chen. There is definition which seems awkward to me.

Def. $V=(X,Y)$ of Bernoulli RVs is said to be Positively Related if there is $V_1=(X_1,Y_1)$ and $V_2=(X_2,Y_2)$ such that $V_1,V_2\ge V$ and $V,V_1$ has the same conditional distribution given $X=1$ and analogously for $V_2,Y$.

The first problem is that I can always set $V_1=V_2=V$ so every vector is positively related!

The second problem is that I'm trying to depict it with Venn diagram. As $X,Y$ are Bernoulli, I can associate a set for each of them. The condition $V_1,V_2 \ge V$ means that the $X\subset X_1,X_2$ and $Y \subset Y_1,Y_2$. The other condition, as I understand is that $X \cap Y_1=X \cap Y=Y \cap X_2$ and doesn't infer anything about positive correlation or something.

What's wrong with my inference?

Thanks in advance. I will be very grateful if you focus on the Venn diagram.

Answer 1

The first problem is that I can always set

No. The distribution of $X$ and the conditional distribution given $Y=1$ are different.

The positive relation says essentially that the distribution of $X$ is dominated by its conditional distribution given $Y=1$ (and similarly for $Y$), which seems quite natural.