Probability distribution of a set of samples from a normal distribution and a cosine function

Question

Let $(x_n)_{n=0}^{N}$ be a finite sequence of samples drawn from a normal distribution with mean $0$ and variance $\sigma^2$. I am interested in the distribution of the sequence $$ X_k = \sum_{n=0}^Nx_n \cos(2 \pi k n /N), \quad k = 0,\dots, N$$ Does this sequence also follow a normal distribution? If so, how can it be proved? It would also be helpful if someone can suggest a reference for this. Thanks.

Answer 1

Yes it does. The question from Signal Processing stack exchange here describes this in detail.

The cosine values are deterministic scalars multiplying the $x_n$ so you get a linear combination of Gaussians which is itself a Gaussian.

Answer 2

If $x_n$, $n=0\ldots N$ are independet, then it follows from elementary properties of characteristic functions of random variables:

Let $\varphi_X(t):=\mathbb{E}e^{itX}$ and $\varphi_Y(t):=\mathbb{E}e^{itY}$, where $X,Y$ are independent random variables. Then $\varphi_{aX+bY}(t)=\varphi_X(at)\varphi_Y(bt)$, $a,b\in\mathbb{R}$.

Additionaly, $\varphi_X=\varphi_Y\iff X,Y$ have the same probability distribution.

Also, it follows from the stability.