Tauberian conditions, under which statistical convergence follows from statistical summability \((C,1)\) (Q1856914)

Let a real or complex sequence \((x_k)\) be given. We say that it converges statistically to some limit \(L\) if for all \(\varepsilon >0\) \[ {1\over n+1} |\bigl\{k\leq n:|x_k-L|\geq \varepsilon \bigr\} |\to 0\quad \text{as}\quad n\to\infty. \] Let \(\sigma_n= \sum^n_{k=0} x_k/ (n+1)\) denote the \((C,1)\)-transform of the sequence. Now, consider a sequence \((x_k)\) such that \((\sigma_n)\) is statistically convergent. The main results in this paper give necessary and sufficient one- and two-sided `statistical' oscillation conditions which imply statistical convergence of the sequence. The results immediately show that `statistical' slow oscillation or `statistical' slow decrease are Tauberian conditions from statistical \((C,1)\)-convergence to statistical convergence. These Tauberian conditions hold under ordinary slow oscillation or slow decrease, which in turn are Tauberian conditions from statistical convergence to ordinary convergence; see \textit{J. A. Fridy} and \textit{M. K. Khan} [Proc. Am. Math. Soc. 128, 2347-2355 (2000; Zbl 0939.40002)].