  Using proof-theoretic methods in the style of proof mining, we give novel
computationally effective limit theorems for the convergence of the
Cesaro-means of certain sequences of random variables. These results are
intimately related to various Strong Laws of Large Numbers and, in that way,
allow for the extraction of quantitative versions of many of these results. In
particular, we produce optimal polynomial bounds in the case of pairwise
independent random variables with uniformly bounded variance, improving on
known results; furthermore, we obtain a new Baum-Katz type result for this
class of random variables. Lastly, we are able to provide a fully quantitative
version of a recent result of Chen and Sung that encompasses many limit
theorems in the Strong Laws of Large Numbers literature.
