Tauberian Theorems via Statistical Convergence

The concept of statistical convergence, which is related to the usual concept of convergence in probability, provides a regular summability method for abstract metric spaces. By using probabilistic tools, we provide some Tauberian theorems which have best possible order Tauberian conditions. Furthermore, these methods can be used to unify and improve the classical pointwise Tauberian theorems of summability theory for the random walk type methods as proved by Bingham, and Hausdorff methods as proved by Lorentz