# Something like Central Limit Theorem for variance and maybe even for covariance?

CLT states in short, that sum/mean of random iid variables from almost any distribution approaches normal distribution.

I failed to find information about asymptotic behavior of sample variance when sample is drawn from unknown distribution. Do we have any reason to believe, that variance of random iid variables asymptotically approach any particular distribution (like chi-squared for normal case)?

What about covariance of multivariate iid distribution? Can we have any reason to believe, that covariance calculated on sample drawn from it can asymptotically approach Wishart distribution? (or any other?)

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See this answer of mine. –  mpiktas Apr 11 '12 at 12:14
This is exactly, what I was looking for. Thanks a lot!! If you make an answer with this reference, I'll accept it. –  Adam Ryczkowski Apr 11 '12 at 14:40
Sample variances and covariances are still averages (for the usual $n-1$ denominator form of variance, it's a constant times an average). So clearly the CLT itself applies. But the chi-square seems to come up at least a little more generally than just for the normal; it appears as an asymptotic result in some nonparametric tests based on variances of ranks as well, for example. –  Glen_b Jun 4 at 23:38