Ygtjki

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
$$(*) , , , mathbbP left[ d_GH[ (m_n,d_n), (X,d) ] < R right] to mathbbP left[ d_GH[ (m_infty,d_infty), (X,d) ] < R right]$$
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?

Following the notation of the paper, let $mathbbK$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $mathrmd_GH$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : mathbbK to mathbbR$, we have $mathbbE[F((m_n, d_n))] to mathbbE[F((m_infty, d_infty))]$. 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 $S = (mathbbK, mathrmd_GH)$, the metric space of all compact metric spaces.

In particular, if $(X,d)$ is a fixed compact metric space, the function $mathrmd_GH(cdot, (X,d)) : mathbbK to mathbbR$ is a continuous function. So if we let $Y_n = mathrmd_GH((m_n, d_n), (X, d))$, 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 $R mapsto mathbbP[mathrmd_GH((m_infty, d_infty), (X,d)) < R]$ is continuous.