# 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?)

• 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
• Muirhead (Aspects of Multivariate Statistical Theory) gives a CLT for the sample covariance matrix of Gaussians (end of section 3.2.2); a statement for more general distributions would involve expressions for the fourth moments (to get the covariances of the covariances). – petrelharp Apr 4 '14 at 21:09

## 1 Answer

For finite populations, as the sample size increases, the variance of the sample variance decreases (the finite population correction). When the sample size is equal to the population size, the sample variance is no longer a random variable. For any finite population, there will not be an asymptotic distribution of the sample variance. See Cochran (1977) Sampling Techniques.

For infinite size populations, see Theorem 5.3.2 on the first page of http://www.unc.edu/~hannig/STOR655/handouts/Handout-asymptotics.pdf

Looks like the sample variance is asymptotically normal!

• Thanks. Do you happen to know is it also true for covariance? – Adam Ryczkowski Apr 11 '12 at 14:43