Statistical extensions of some classical Tauberian theorems

Hardy's well-known Tauberian theorem for Cesàro means says that if the sequence $x$ satisfies $\lim Cx = L$ and $\Delta x_k = O (1/k)$, then $\lim x = L$. In this paper it is shown that the hypothesis $\lim Cx = L$ can be replaced by the weaker assumption of the statistical limit: st-lim $Cx = L$, i.e., for every $\epsilon >0$, $\lim n^{-1} | \{ k \leq n: | ( Cx)_k - L | \geq \epsilon \} | = 0$. Similarly, the ``one-sided'' Tauberian theorem of Landau and Schmidt's Tauberian theorem for the Abel method are extended by replacing $\lim Cx$ and $\lim Ax$ with st-lim $Cx$ and st-lim $Ax$, respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing $\lim _t (Bx)_t=L$, where $t$ is a continuous parameter, with $\lim _n (Bx)_n =L$, and further replacing it by $(B^{*})$-st-lim $B^{*} x =L$, where $B^{*}$ is the Borel matrix method.