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.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.