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I have a centered Gaussian sample of $n$ elements $X_i,\,i=1,..,n$, with variance $\sigma^2$. I would like to find the limiting distribution of the sample variance $\sigma_n^2=\frac 1n \sum_{i=1}^n X_i^2$ and of $\sqrt n(\sigma_n-\sigma)$, using the Central Limit Theorem. Thank you.

kjetil b halvorsen
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marco
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  • This has been asked and answered in this site, here is one answer https://stats.stackexchange.com/questions/105337/asymptotic-distribution-of-sample-variance-of-non-normal-sample/203697#203697. Do you know about the delta method? – JohnK Jan 09 '20 at 18:39
  • Actually, in the post you pointed to, it seems that they do not obtain the limit distribution of $\sqrt n (\sigma_n - \sigma)$, but of the same expression valid for the variance. Did I miss anything? – marco Jan 09 '20 at 18:43

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