Limit of sequence of characteristic functions with Riemann integral approximation

Question

Given the following sequence of characteristic functions $\phi_{X_{n}}(t)=\frac{1}{n}\sum_{k=1}^{n}\exp(2it\cos(2\pi \frac{k}{n}))$ where $ 1 \le k \le n$ does it converge to the characteristic function of some random variable? I thought about approximating it with an integral from the definition of the Riemann integral.

Answer 1

$\phi_{X_n}(t)$ is a Riemann sum for $\int_0^1 \exp(2it\cos(2\pi x) dx$, to which it converges because the integrand is continuous on $[0,1]$. The limit random variable will have the same distribution as $X=2\cos(2\pi U)$, where $U$ is uniform on $[0,1]$. The r.v. $X$ takes values in $[-2,2]$, and it's not difficult to find its density function.