Summary: The first named author has recently proved necessary and sufficient Tauberian conditions under which statistical convergence follows from statistical summability . The aim of the present paper is to generalize these results to a large class of summability methods by weighted means.
Let be a sequence of nonnegative numbers such that and
Let be a sequence of real or complex numbers and set for . We present necessary and sufficient conditions under which the existence of the limit follows that of , where is a finite number. If is a sequence of real numbers, then these are one-sided Tauberian conditions. If is a sequence of complex numbers, then these are two-sided Tauberian conditions.