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Tauberian conditions, under which statistical convergence follows from statistical summability $\left(C,1\right)$. (English) Zbl 1021.40002

Let a real or complex sequence $\left({x}_{k}\right)$ be given. We say that it converges statistically to some limit $L$ if for all $\epsilon >0$

$\frac{1}{n+1}|\left\{k\le n:|{x}_{k}-L|\ge \epsilon \right\}|\to 0\phantom{\rule{1.em}{0ex}}\text{as}\phantom{\rule{1.em}{0ex}}n\to \infty ·$

Let ${\sigma }_{n}={\sum }_{k=0}^{n}{x}_{k}/\left(n+1\right)$ denote the $\left(C,1\right)$-transform of the sequence. Now, consider a sequence $\left({x}_{k}\right)$ such that $\left({\sigma }_{n}\right)$ 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 $\left(C,1\right)$-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 J. A. Fridy and M. K. Khan [Proc. Am. Math. Soc. 128, 2347-2355 (2000; Zbl 0939.40002)].

##### MSC:
 40E05 Tauberian theorems, general 40G05 Cesàro, Euler, Nörlund and Hausdorff methods
Cesàro means