Are these two random variables $B_1$ and $B_2$ independent of each other?

Question

$\{\Phi_n\}$ are independent and identically distributed random variables distributed uniformly on $[0,2\pi]$ and $$B_1 = \frac{1}{\sqrt{N}}\sum_{n=1}^N \cos\Phi_n$$ $$B_2 = \frac{1}{\sqrt{N}}\sum_{n=1}^N \sin\Phi_n,$$ and I want to prove that when $N$ tends to infinity, $B_1$ and $B_2$ are independent of each other. I know this does not sound like a rigorous statement, but I don't know how to put it rigorously.

I think maybe the central limit theorem can be applied here, but I am not sure how.

Answer 1

Comment: I can summon very little intuition about this problem. So I tried a straightforward simulation in R statistical software.

From a simulation of 100,000 iterations, each with n = 20, one can see that the sample correlation of $B_1$ and $B_2$ is consistent with 0 population correlation. Furthermore, no association is evident in the scatterplot.

m = 10^5;  n = 20
phi = runif(m*n, 0, 2*pi)
PHI = matrix(phi, nrow=m)
A.1 = rowSums(sin(PHI))
A.2 = rowSums(cos(PHI))
B.1 = A.1/sqrt(n); B.2 = A.2/sqrt(n)
cor(B.1, B.2)
## -0.00383627

Note: This is a corrected post. The original had typos in the R code. I regret any inconvenience from posting garbage for an hour.