Abstract. In this paper we provide various approximation results concerning the classical Korovkin theorem via A -statistical convergence. We also study the rates of A -statistical convergence of approximating positive linear operators and give some examples.

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. ᮊ 1998 Academic Press

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The main objective of the paper is to characterize multipliers of summability fields of regular methods, while relaxing the usual boundedness condition. For this purpose we use the sequence spaces of A-bounded sequences and A-uniformly integrable sequences. Among the main results, it is shown that the space of multipliers is closely related to the space of A-statistically convergent sequences and that A-statistical convergence over ∞ is equivalent to a regular matrix method. This observation eliminates the need for separate proofs of several A-statistical convergence results.