Solved – 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 $n rightarrow infty$ ?

A. van der Vaart gives derivation of assymptotic distribution of sample variance in his book as an example of delta method. It would be not that hard (and very instructive) to adapt the derivation for the case of standard deviation.

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