Weak convergence of couplings of probability measures

Question

I have the following problem:

I am given a Polish space $(\mathcal{X},d)$ and a weakly convergent sequence of probability measures $ (\mu_k)_{k\in\mathbb{N}} $ with limit $\tilde{\mu}$. Given another probability measure $\mu$ and a sequence $(\pi_k)_{k\in\mathbb{N}}$ of couplings of $(\mu_k,\mu)$.

My question: Does $(\pi_k)_{k\in\mathbb{N}}$ converge to some coupling $\pi$? I was trying to prove the marginal property of $\pi$, but couldn't quite figure it out. Is it even guaranteed that $\pi_k$ converges at all?

Thanks a lot

Answer 1

Using Prokhorov's theorem and the fact that $\left(\mathcal X,d\right)$ is Polish, it can be shown that the sequence $\left(\mu_k\right)_{k\in\mathbb N}$ is tight hence so is the sequence $\left(\pi_k\right)_{ k\in\mathbb N}$. Therefore, the latter sequence admits a convergent subsequence. The potential limit is a coupling of $\left(\widetilde{\mu},\mu\right)$, but $\left(\pi_k\right)_{ k\in\mathbb N}$ may not be convergent in general since coupling are not unique in general.