Statistical extensions of some classical Tauberian theorems

Source abstract

Hardy’s well-known Tauberian theorem for Cesàro means says that if the sequence x x satisfies lim C x = L \lim Cx = L and Δ x k = O ( 1 / k ) \Delta x_k = O (1/k) , then lim x = L \lim x = L . In this paper it is shown that the hypothesis lim C x = L \lim Cx = L can be replaced by the weaker assumption of the statistical limit: st-lim C x = L Cx = L , i.e., for every ϵ > 0 \epsilon >0 , lim n − 1 | { k ≤ n : | ( C x ) k − L | ≥ ϵ } | = 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 C x \lim Cx and lim A x \lim Ax with st-lim C x Cx and st-lim A x Ax , respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing lim t ( B x ) t = L \lim _t (Bx)_t=L , where t t is a continuous parameter, with lim n ( B x ) n = L \lim _n (Bx)_n =L , and further replacing it by ( B ∗ ) (B^{*}) -st-lim B ∗ x = L B^{*} x =L , where B ∗ B^{*} is the Borel matrix method.