My friend and I have a problem on central limit theorem.
Given $X_1,X_2......$ are i.i.d random variables with mean $\mu$=0, variance $\sigma^2=1$(may or may not be normally distributed). If we define: $Y_n=X_1X_2+X_2X_3+....X_{n-1}X_n$, how to show the $\frac{Y_n}{n}\to N(0,1)$ when $n\to\infty$. i.e. $$\frac{X_1X_2+X_2X_3+....X_{n-1}X_n}{\sqrt{n}}\to N(0,1),as \ n\to \infty$$
I am not sure denominator is n or n-1. It is obvious that $X_1X_2$ and $X_2X_3 $ are dependent, so I cannot use the standard central limit theorem. I also find the $\textit{Lindeberg-Feller CLT}$, but it is on the independent but not identically distributed random variables.
Thanks a lot for help!