What does convergence in distribution "in the Gromov–Hausdorff" sense mean?

Question

I am trying to understand this survey article by Le Gall on Brownian geometry, especially the statement of Theorem 1.

The basic statement of the theorem is $$(m_n,d_n) \to (m_{\infty}, d_{\infty})$$ "in the Gromov–Hausdorff sense" as $n \to \infty$, where the convergence is in distribution.

Here $(m_n,d_n)$ and $(m_{\infty},d_{\infty})$ are both random compact metric spaces. So how do we interpret this? We might hope for a statement along the lines of the following.

For every compact metric space $(X,d)$ and $R > 0$, we have $$(*) \, \, \, \mathbb{P} \left[ d_{GH}[ (m_n,d_n), (X,d) ] < R \right] \to \mathbb{P} \left[ d_{GH}[ (m_{\infty},d_{\infty}), (X,d) ] < R \right]$$ as $n \to \infty$.

But even when we talk about convergence in distribution for real random variables (instead of compact-metric-space-valued random variables), we have to be careful to restrict our attention to points where the cumulative distribution function is continuous. So I wonder if (*) is too strong?

Answer 1

Following the notation of the paper, let $\mathbb{K}$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $\mathrm{d_{GH}}$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : \mathbb{K} \to \mathbb{R}$, we have $\mathbb{E}[F((m_n, d_n))] \to \mathbb{E}[F((m_\infty, d_\infty))]$. The portmanteau theorem gives you several other equivalent statements.

In other words, this is just the usual notion of convergence in distribution for random variables taking their values in a metric space $S$, where that metric space happens to be $S = (\mathbb{K}, \mathrm{d_{GH}})$, the metric space of all compact metric spaces.

In particular, if $(X,d)$ is a fixed compact metric space, the function $\mathrm{d_{GH}}(\cdot, (X,d)) : \mathbb{K} \to \mathbb{R}$ is a continuous function. So if we let $Y_n = \mathrm{d_{GH}}((m_n, d_n), (X, d))$, then the scalar-valued random variables $Y_n$ converge in distribution to $Y$. So your formula (*) holds, but as you say, only for values of $R$ at which the function $R \mapsto \mathbb{P}[\mathrm{d_{GH}}((m_\infty, d_\infty), (X,d)) < R]$ is continuous.