Method of Moments for an operator in quantum random walk

I was reading a paper by Grimmett, Janson, and Scudo, see here for arxiv version of the paper, where they show that for all positive integers

$$\lim_{n \to \infty}E[(X_{n}/n)^{r}] =\int_{\Omega} h^{r} d\mu$$

where $$h$$ is a bounded continous function (this is equation (19) in the paper), then say that $$X_{n}/n \to h$$ weakly by the method of moments and the fact that $$h$$ is bounded. Normally, if these $$X_{n}$$ were random variables, then I think this is a consequence of the Levy's continuity theorem, but in this case, we don't have random variables but we have that moments here are respect to the operator $$D = -i\frac{d}{dk}$$, that is we have that $$X_{n} = (U^{*})^{n}DU^{n}$$ and $$r$$th moments with respect to some initial state $$\psi_{0} \in L^{2}(\mathbb{K}) \otimes \mathbb{C}^{2}$$ (where $$\mathbb{K}$$ is the unit circle) defined as $$\int \langle U^{n}\psi_{0}, D^{r}U^{n}\psi_{0} \rangle \frac{dk}{2\pi}$$

To this end, I'm confused on two points, namely, what does weak convergence mean here with respect this and why does convergence in moment imply weak convergence.

Here is a screenshot of the proof, I'm confused about for ease of reference: