Central limit theorem for uncorrelated identically distributed random variable

I have a sum of random variables as bellow $$Y=\sum_n A_n=\sum_n B_n\times C_n$$ where $B_n$s are correlated Gaussian random variables with zero mean, variance $1$ and correlation $E\{B_nB^*_r\}=\frac{1}{N}\sum_{l-0}^{L-1}\sigma^2_le^{\frac{j2\pi l(n-r)}{N}}$ where $\sum_{l=0}^{L-1}\sigma^2_l=1$ and $L$ and $N$ are integers ($L<N$). $C_n$s are independent and identically distributed random variables with zero mean and variance $P$. Also $B_n$ is independent of $C_n$. My question is that can the distribution of $Y$ be approximated as Gaussian? I know that the central limit theorem is true for sum of iid random variables but $Y$ is sum of uncorrelated identically distributed random variable. I can't prove whether $A_n$s are independent or not. Does the central limit theorem hold for my case?

• For $normal$ data, uncorrelated implies independent. But your situation involves quantities that are presumably $converging$ to normal. – BruceET Feb 13 '16 at 20:35