In CLT, we require each $X_{n}$ to be independent and identically distributed. Then we have $\frac{Sn}{\sqrt{n}}$ converges weakly to $N(0,1)$.
But if we have the sequence ${Xn}$ to be standard normal distribution but not necessarily independent. Then what happens to $\frac{Sn}{\sqrt{n}}$?
I know that it can converge to many different values, but what are the possible outcomes? Is it possible to converge to all normal distributions with mean $0$, for example to $N(0,2)$?
My idea is to use the characteristic function. We have $E_{X}(e^{it\sqrt{n}X})$, where $X$ stands for standard normal distribution $N(0,1)$. Then we should compare it with characteristic functions of other normal distributions but I've no idea how to do it