Statistical extensions of some classical Tauberian theorems

Hardy’s well-known Tauberian theorem for Cesàro means says that if the sequence x satisfies limCx = L and ∆xk = O(1/k), then limx = L. In this paper it is shown that the hypothesis limCx = L can be replaced by the weaker assumption of the statistical limit: st-lim Cx = L, i.e., for every > 0, limn−1|{k ≤ n : |(Cx)k − L| ≥ }| = 0. Similarly, the “one-sided” Tauberian theorem of Landau and Schmidt’s Tauberian theorem for the Abel method are extended by replacing limCx and limAx with st-lim Cx and st-lim Ax, respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing limt(Bx)t = L, where t is a continuous parameter, with limn(Bx)n = L, and further replacing it by (B∗)-st-lim B∗x = L, where B∗ is the Borel matrix method.