Statistical extensions of some classical Tauberian theorems

Hardy's well-known Tauberian theorem for Cesaro means says that if the sequ ence x satisfies lim Cx = L and Delta x(k) = O(1/k), then lim x = L. In thi s paper it is shown that the hypothesis lim Cx = L can be replaced by the w eaker assumption of the statistical limit: st-lim Cx = L, i.e., for every e psilon >0, lim n(-1)\{k less than or equal to n : \(Cx)(k) - L\ greater tha n or equal to epsilon}\ = 0. Similarly, the "one-sided" Tauberian theorem o f Landau and Schmidt's Tauberian theorem for the Abel method are extended b y replacing lim Cx and lim Ax with st-lim Cx and st-lim Ax, respectively. T he Hardy-Littlewood Tauberian theorem for Borel summability is also extende d by replacing lim(t)(Bx)(t) = L, where t is a continuous parameter, with l im(n)(Bx)(n) = L, and further replacing it by (B*)-st-lim B*x = L, where B* is the Borel matrix method.