zbMATH — the first resource for mathematics

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.

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