Convergence of sum of random variables to the normal distribution

Question

The classical CLT speaks about the convergence of centered and normed sum of random variables to the standard normal variable. But what about the convergence of noncentered and nonnormed sum of r.v.s'? Im my case I have that $X_k\sim N(0,\frac{C}{n})$ (where $C>0$, $n$ is an integer) and then I need a limit of the distribution of the following sum: $$S_n=n\sum_{k=1}^{n} X_k^3$$ The expectation of $S_N$ is $0$, variance is $n^2\sum_{k=1}^{n}\frac{15C^3}{n^3}=15C^2$. So the variance is always constant. working with the distribution function of $X^3$ is not practical - no closed expressions can be obtained for the characteristic function. In case of centered and normed $S_n$ CLT provides a limit. But is it true that the sum $S_n$ itself converge to the normal distribution with parameters $(0,15C^2)$? The computer simulation studies shows that for large $n$ this is a valid approximation. But maybe it isn't true in the limit?