On a Family of Summability Methods and One-Sided Tauberian Conditions

We discuss a large family of summability methods including power series methods, generalized Nörlund means, running means and investigate Abelian and Tauberian relations in this family. The main goal is to find weakest possible one-sided Tauberian conditions for the Tauberian conclusions. From these results we can derive strong laws and Erdős-Rényi laws for all the methods in question when applied to i.i.d. random variables.