# Distribution of standard deviation

This question addressed the normal distribution, but I am wondering what is known about the distribution of the standard deviation of a sample of size n drawn from an arbitrary distribution. In particular, what is the standard deviation of the standard deviation?

For a normal distribution, the s.d. of the s.d. is $\sigma \over{\sqrt{2n}}$. Is this approximately true for an arbitrary distribution as $n \rightarrow \infty$ ?