19
$\begingroup$

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.

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.