Probability Theory — Coupling

A coupling of random variables (which potentially live on different probability spaces) is a way to relate random variables.

For instance, take $𝑋\sim Bernoulli(𝑝)$ and $𝑌\sim Bernoulli(𝑞)$. We have described the random variables, but we have not described the relationship between them. Even if we specify $𝑝<𝑞$, ambiguity remains about the relationship between these two random variables.

To couple random variables, we create a shared probability space, and a random variable who’s marginals are equivalent to the original random variables. The joint distribution on the shared probability space is the coupling.

Example 1 (independent coupling)

Let $𝑋\stackrel{𝑑}{=}{𝑋}^{\prime }$ and $𝑌\stackrel{𝑑}{=}{𝑌}^{\prime }$, where ${𝑋}^{\prime }⫫{𝑌}^{\prime }$.

The random variable $({𝑋}^{\prime },{𝑌}^{\prime })$ has a joint distribution on a new probability space

The marginal distributions of the joint distribution match those of $𝑋$ and $𝑌$ — one can check by summing across columns and rows respectively. By creating this joint distribution (this coupling), we have described a relationship between $𝑋$ and $𝑌$ which can be used to ask questions about the relationship between them. But this is not the only such coupling. $∎$

Example 2 (monotone coupling)

Let $𝑈\sim 𝑈(0,1)$ a uniform random variable.

Now define ${𝑋}^{″}={𝕀}_{\{𝑈\le 𝑝\}}$ and ${𝑌}^{″}={𝕀}_{\{𝑈\le 𝑞\}}$. We can define a new coupling between $𝑋$ and $𝑌$ in terms of these random variables:

The marginal distributions again agree with $𝑋$ and $𝑌$. So we have another, different relationship between the random variables $𝑋$ and $𝑌$. This again can be used to answer questions about their relationship. $∎$

So we see that couplings are useful when we want to describe the relationship between two random variables. Simply describing the distribution of random variables to one another is not sufficient to answer questions like $\mathbb{P}[𝑋=𝑌]$, etc.