Quantitative Strong Laws of Large Numbers

TL;DR

This paper applies proof-theoretic methods to derive effective, quantitative versions of classical strong laws of large numbers, providing optimal bounds and new results for specific classes of random variables.

Contribution

It introduces novel computationally effective limit theorems for Cesaro-means, improving bounds and extending results in the theory of strong laws of large numbers.

Findings

Optimal polynomial bounds for pairwise independent variables with bounded variance.

A new Baum-Katz type result for this class of variables.

A fully quantitative version of a recent theorem by Chen and Sung.

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.