Quantitative Strong Laws of Large Numbers

Abstract: 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.