Let $X$ be uniformly distributed on $[0,1]$ and consider the sequence of functions $f_n(X) = \sin(2n\pi X)$. Let $\phi:\Omega\rightarrow [0,2\pi]$ be a uniformly distributed random variable. Let $\phi_n=\phi(\omega_n)$ for $\omega_n$ all distinct. Then apparently, for large $N$, $(2/N)^{1/2}\sum_{n=1}^N f_n(X-\phi_n)$ will then be approximately normally distributed with mean zero and variance one for almost all realizations of the phases. Yet as soon as $\phi_n$, $n\leq N$, are realized, $f_n(X-\phi_n)$, $n\leq N$, are not independent random variables and yet they still obey the conclusion of the central limit theorem. Why?
I want to make the following conjecture: The probability distribution function of $s_{N;\phi_1,\dots, \phi_N}(X) = (2/N)^{1/2}\sum_{n=1}^N f_n(X-\phi_n)$ is asymptotic to the standard normal distribution function as $N\rightarrow \infty$ for almost all vectors $(\phi_1,\dots,\phi_N)\in[0,2\pi]^N$, the sequence being interpreted as $(\phi_1), (\phi_1, \phi_2),...(\phi_1,\phi_2,...,\phi_N),...$. Is this sensible?
If it is true and the proof is trivial, I would appreciates hints on how to show it more so than the full proof.