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On almost convergent and statistically convergent subsequences. (English) Zbl 0989.40002
A bounded sequence $$s=(s_{n})$$ is almost convergent to $$L$$ if $\lim_{k}\frac{1}{k}\sum_{i=0}^{n-1}s_{n+i}=L,\quad \text{uniformly in }n .$ We write $$f$$-$$\lim s=L$$ and $$\mathbf F=\{s=(s_{n}): f\text{-}\lim s=L\text{ for some }L\}.$$ The sequence $$s=(s_{n})$$ is called statistically convergent to $$L$$ provided that $$\lim_{n}n^{-1}\left|\left\{k\leq n:\left|s_{k}-L\right|\geq \varepsilon \right\} \right|=0$$, for each $$\varepsilon>0,$$ where the vertical bars indicate the number of elements in the enclosed set. We write st-$$\lim s=L$$ and $$\mathbf S=\{s=(s_{n}):\text{st-}\lim s=L \text{ for some }L\}.$$ The authors prove that $$\mathbf F\nsubseteqq \mathbf S$$ and $$\mathbf S\nsubseteqq \mathbf {F}$$. They also examine the Lebesgue measure and the Baire category of the set of all almost convergent (respectively statistically convergent) subsequences of a given sequence.

##### MSC:
 40A05 Convergence and divergence of series and sequences 40D25 Inclusion and equivalence theorems in summability theory 28A12 Contents, measures, outer measures, capacities
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