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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:

enter image description here

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