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?)