Coupling of vectors

Question

Let $X = (X_1,X_2)$ and $\hat X = (\hat X_1,\hat X_2)$ be two random variables where $X_i,\hat X_i$ are taking values over the Polish space $E_i$ endowed with their Borel $\sigma$-algebras, where $i=1,2$.

Let $X_1$ has a distribution $\mu$ and $\hat X_1$ has a distribution $\hat \mu$. Furthermore, suppose that $K$ is a conditional kernel on $E_2$ given $E_1$ which describe the distribution of $X_2$ given $X_1$, i.e. $$ P(X_2\in B|X_1) = K_{X_1}(B) $$ where $B\in \mathfrak B(E_2)$ is any Borel measurable set and $\mathsf P$ is a joint distribution of $X = (X_1,X_2)$. Let $\hat K$ be defined similarly for $\hat X_2$ given $\hat X_1$ and let $\hat P$ be the joint distribution of $\hat X = (\hat X_1,\hat X_2)$.

Suppose that for the total variation it holds that $\|P - \hat P\|>0$. Let $E = E_1\times E_2$ be the product space. The measure $\Bbb P$ on $E^2$ is called a coupling of $\mathsf P$ and $\hat {\mathsf P}$ if $$ \Bbb P\circ\pi^{-1} = \mathsf P,\quad \Bbb P\circ\hat \pi^{-1} = \hat{\mathsf P} $$ where $\pi,\hat \pi$ are the corresponding projections maps. The coupling $\Bbb P$ is called maximal if

1. it holds that $$\|P - \hat P\| = 2\Bbb P(X\neq \hat X)\tag{1} $$

2. $X$ and $X'$ are $\Bbb P$-independent conditional on $\{X\neq \hat X\}$, i.e. $$ \Bbb P(X\in A,\hat X\in \hat A|X\neq \hat X) = \Bbb P(X\in A|X\neq \hat X)\Bbb P(\hat X\in \hat A|X\neq \hat X) $$ for any sets$A,\hat A\in \mathrm B(E)$.

The maximal coupling always exists and is unique.

I have two questions:

• is that true that the maximal coupling of $X$ and $\hat X$ is also a maximal coupling of their coordinates $X_1, \hat X_1$ and $X_2, \hat X_2$? Here I mean the projection of $\Bbb P$ on the correspondent spaces. Or at least, does $(1)$ holds for the projected coupling measures?

• since the maximal coupling $\Bbb P$ is unique, can you suggest how to express $\Bbb P(X = \hat X)$ in terms of $\mu,\hat \mu$ and $K,\hat K$?

I am not experienced in conditioning, so any help is appreciated. I know that $$ \Bbb P(X = \hat X) = \Bbb P(X_1 = \hat X_1)\Bbb P(X_2 = \hat X_2|X_1 = \hat X_1) $$ but I am not sure even how to compute the first term in the RHS.

Answer 1

1. No, this is not true. For example, let $E_1=E_2=\{0,1\}$, let $P$ be the uniform distribution on $\{(0,0),(1,1)\}$ and let $\hat P$ be the uniform distribution on $\{(0,1),(1,0)\}$. By (2) the maximal coupling $\mathbb P$ is the product distribution $P\otimes\hat P$, so the pushforward measure on $E_1\times E_1$ (or $E_2\times E_2$) is the uniform distribution on $\{0,1\}^2$, which is never a maximal coupling.

2. The question already mentions how $\mathbb P(X=\hat X)$ is related to the total variation distance of $P$ and $\hat P$. And $P$ is given in terms of $K$. To make the dependence on $\mu$ more explicit, how about: $$P(B)=\int_{x\in E_1} K_{x}(\{y\in E_2\colon (x,y)\in B\}) d\mu(x)\qquad (B\in\mathcal B(E_1\times E_2)).$$