# Intuition behind Strassen's theorem

Currently I am dissecting a proof of Strassen's theorem, which states the following:

Suppose that $(X,d)$ is a separable metric space and that $\alpha,\beta>0$. If $\mathbb{P}$ and $\mathbb{Q}$ are probability measures on $X$ satisfying $$\mathbb{P}(E) \le \mathbb{Q}(E^\alpha) + \beta$$ for any Borel-measurable set $E \subset X$, then for any $\varepsilon>0$ there exist two nonnegative measures $\mu,\nu$ on $X\times X$ such that

1. $\mu + \nu$ is a law on $X\times X$ with marginals $\mathbb{P}$ and $\mathbb{Q}$.

2. $\mu \{(x,y)\in X\times X: d(x,y) > \alpha + \varepsilon \} = 0$

3. $\nu(X\times X) \le \beta + \varepsilon$

Here $X$ is endowed with the Borel $\sigma$-algebra arising from the metric topology, and $E^\alpha = \{x \in X : d(x,y) < \alpha \text{ for some } y \in E\}$ is the $\varepsilon$-enlargement of $E$.

Right now I'm mainly concerned about the intuition behind this theorem and how it relates to things I've recently learned (say, the Prokhorov and Ky-Fan metrics and various metrizations of convergences we care about when dealing with random variables.)

My understanding of it so far is that, if we have a generalized notion of the Prokhorov distance constraining $\mathbb{P}$ in terms of $\mathbb{Q}$, then we can always find a probability measure on the product space that in some sense "couples" $\mathbb{P}$ and $\mathbb{Q}$ and does so in a way that concentrates mass close to the diagonal. Admittedly though, this is a stretch of the imagination -- would more experienced probabilists please elucidate this result? I'd also love to know specific places where one might find occasion to use this.

## 1 Answer

Well, certainly it is a useful result for coupling. Let us define $M_\epsilon:=\mu+\nu$, then $M_\epsilon$ is a coupling of $\Bbb P$ and $\Bbb Q$ by $(1)$, and $$M\left(d(x,y) \geq \alpha+\epsilon\right)\leq\beta+\epsilon.$$ The latter fact is very useful: say, you have two random variables $\xi\sim \Bbb P$ and $\eta\sim \Bbb Q$, and you would like to use $\eta$ to approximate certain properties of $\xi$. For example, $\xi$ is a nasty stochastic process and $\eta$ is its discretization. Suppose that you know that $\Bbb Q(\eta_t \in A \;\forall t\geq T) = 1$ for some attractor set $A$; how can you use this information to argue about the limiting behavior of $\xi$? So far you are only given their distributions, so you want to connect them somehow. In case $d(x,y) = \sup_t |x_t - y_u|$, the result in the OP tells you that $$\Bbb P(\xi_t\in A_{\alpha+\epsilon} \; \forall t\geq T)\geq 1 - \beta-\epsilon$$ where $A_{\alpha+\epsilon}$ is a ball of an obvious radius around $A$, which maybe quite useful in applications. Quite similar estimates can be derived using the Wasserstein metric. See the discussion on p.3 here.