# What do we know about the rate of convergence of the mean of RVs with infinite variance?

What, if anything, do we know about the rate of convergence of of the mean of identically distributed, independent or stationary random variables drawn from a distribution with a finite mean and infinite or nonexistent variance?

I have asked a separate question about the conditions under which the mean of such variables do converge. In answering this question, you may assume that such conditions, whatever they are, hold.