9
$\begingroup$

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?

$\endgroup$
11
$\begingroup$

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.

$\endgroup$

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.