# Weak convergence of $\frac{1}{n} \sum_{i = 1}^{n} \delta_{2 \cos(2 \pi k / n)}$ to $\frac{1}{\pi \sqrt{4 - x^2}} {\large\chi}_{|x|\leq 2} dx,$

My questions come from the pdf https://www.math.univ-toulouse.fr/~bordenave/coursSRG.pdf that I am reading. More specifically, on page 6, the author computes the spectral measure for finite graphs, and in particular for $$C_n$$ a cycle of length $$n$$.

If $$G = (V, E)$$ is a finite graph, $$|V| = n$$, the author defines the spectral measure of $$G$$ as

## $$\mu_G = \frac{1}{n} \sum_{i = 1}^{n} \delta_{\lambda_i}$$

where $$(\lambda_1, \ldots, \lambda_n)$$ are the eigenvalues (counting multiplicity) of $$A$$, the adjacency operator.

Let $$C_n$$ be a cycle of length $$n$$. The adjacency operator can be written as $$A = B + B^\ast$$ where $$B$$ is the permutation matrix of a cycle of length $$n$$. Since $$BB^\ast = B^\ast B = I$$, the eigenvalues of $$B$$ are the roots of unity and the eigenvalues of $$A$$ are $$\lambda_k = 2 \cos(2\pi k /n)$$, $$1 \le k \le n$$. We get

## $$\mu_{C_n} = \frac{1}{n} \sum_{i = 1}^{n} \delta_{2 \cos(2 \pi k / n)}$$

As $$n$$ goes to infinity, $$\mu_{C_n}$$ converges weakly to a arcsine distribution $$\nu$$ with density on $$[-2,2]$$ given by

## $$d \nu(x) = \frac{1}{\pi \sqrt{4 - x^2}} {\large\chi}_{|x|\leq 2} dx,$$

($$\nu$$ is the law of $$2 \cos(\pi U)$$ with $$U$$ uniform on [0,1]).

### Question

I am not able to prove the weak convergence asserted by the author. Is someone able to prove it/elaborate on why this is true?

Let $$X_n$$ be the discrete random variable that attains the value $$k/n$$ with probability $$1/n$$, for $$k=1,\dotsc,n$$. The sequence $$(X_n)_n$$ converges in distribution to $$U\sim\mathrm{Uniform}(0,1)$$.
Consider the continuous function $$g(x)=2\cos(2\pi x)$$. By the properties of convergence in distribution, namely the continuous mapping theorem, the variables $$g(X_n)$$ converge in distribution to $$g(U)$$.