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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 $n \rightarrow \infty$ ?

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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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If the distribution has a mean and standard deviation, then I'd say yes.

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    $\begingroup$ We look for replies that provide support or at least references. What are yours? $\endgroup$
    – whuber ♦
    Commented Aug 1, 2011 at 2:25
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    $\begingroup$ given that in the large sample limit, the central theorem applies, then I believe the suggested result holds. Can't be more of a help though... $\endgroup$
    – yannick
    Commented Aug 1, 2011 at 6:01
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    $\begingroup$ Thanks; that's a good start. There may be difficulties with distributions without fourth moments, though. $\endgroup$
    – whuber ♦
    Commented Aug 1, 2011 at 12:39

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