This post is inspired by an assignment question I had to answer for STATS 310A – a probability course at Stanford for first year students in the statistics PhD program. In the question we had to derive a few results about couplings. I found myself thinking and talking about the question long after submitting the assignment and decided to put my thoughts on paper. I would like to thank our lecturer Prof. Diaconis for answering my questions and pointing me in the right direction.

## What are couplings?

Given two distribution functions and on , a ** coupling** of and is a distribution function on such that the marginals of are and . Couplings can be used to give probabilistic proofs of analytic statements about and (see here). Couplings are also are studied in their own right in the theory optimal transport.

We can think of and as being the cumulative distribution functions of some random variables and . A coupling of and thus corresponds to a random vector where has the same distribution as , has the same distribution as and .

## The independent coupling

For two given distributions function and there exist many possible couplings. For example we could take where . This coupling corresponds to a random vector where and are independent and (as is required for all couplings) , .

In some sense the coupling is in the “middle” of all couplings. This is because and are independent and so doesn’t carry any information about . As the title of the post suggests, there are couplings were this isn’t the case and carries “as much information as possible” about .

## The two extremal couplings

Define two function by

and .

With some work, one can show that and are distributions functions on and that they have the correct marginals. In this post I would like to talk about how to construct random vectors and .

Let and be the quantile functions of and . That is,

and .

Now let be a random variable that is uniformly distributed on and define

and .

Since if and only if , we have and likewise . Furthermore occurs if and only if which is equivalent to . Thus

Thus is distributed according to . We see that under the coupling , and are closely related as they are both increasing functions of a common random variable .

We can follow a similar construction for . Define

and .

Thus and are again functions of a common random variable but is an increasing function of and is a decreasing function of . Note that is also uniformly distributed on . Thus and .

Now occurs if and only if and which occurs if and only if . If , then and . On the other hand, if , then and . Thus

,

and so is distributed according to .

## What makes and extreme?

Now that we know that and are indeed couplings, it is natural to ask what makes them “extreme”. What we would like to say is that is an increasing function of and is a decreasing function of . Unfortunately this isn’t always the case as can be seen by taking to be constant and to be continuous.

However the intuition that is increasing in and is decreasing in is close to correct. Given a coupling , we can look at the quantity

This quantity tells us something about how changes with . For instance if and were positively correlated, then would be positive and if and were negatively correlated, then would be negative.

For the independent coupling , the quantity is constantly . It turns out that the above probability is maximised by the coupling and minimised by and it is in this sense that they are extremal. This final claim is the two dimensional version of the Fréchet-Hoeffding Theorem and checking it is a good exercise.

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