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Given two Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}$, let $\Pi(\mu, \nu)$ denote all couplings between them, i.e., all Borel probability measures on $\mathbb{R}^2$ such that the marginal distribution of the first and second coordinate are $\mu$ and $\nu$ respectively. Can we describe the set of all possible support of the joint law:

$S(\mu, \nu) = \{E \subset \mathbb{R}^2: \exists \lambda \in \Pi(\mu, \nu), s.t. \lambda(E) = 1\}$.

Strassen (Theorem 11) characterized this collection: $C \in S(\mu, \nu)$ if and only if for all $U$ open in $\mathbb{R}$, $\nu(U) \leq \mu(U^C)$, where $U^C = \{x: \exists y \in U, s.t. (x,y) \in C\}$. But this condition seems not easy to verify! For instance, it is probably not true to only consider open interval $U$.

Let us focus on the normal distribution $\mu=\nu=N(0,1)$. Can we characterize $S' = S(N(0,1), N(0,1))$ more explicitly in this case, i.e., what kind of subset in the plane admits a probability measure whose marginals are standard normal? For instance we can ask the following question: for what pair of $a < b$ do we have $\{(x,y): a \leq |x-y| \leq b\} \in S'$, that is, can we construct a joint distribution on this strip that has standard normal marginals. It seems a non-trivial question. For $b = \infty$, one can show that the largest possible $a$ is between $\sqrt{\pi/2}$ and $3/2$. I am thinking is there a systematic way to do it in the simple case of real line. Note that Strassen's result work for any Polish space.

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