On statistical limit points. (English) Zbl 0966.40001
A real number \(x\) is said to be a statistical limit point of the sequence \(x_{n}\) if there exists a subsequence \(x_{k_{n}}\), \(n=1,2,\dots\), such that \(\lim_{n \to \infty} x_{k_{n}}=x\) and the set of indices \(k_{n}\) has a positive upper asymptotic density. The paper gives a characterization of the set of all statistical limit points of a given sequence \(x_{n}\) by means of \(F_{\sigma}\) sets.
Reviewer: Vasile Berinde (Baia Mare)
MSC:
| 40A05 | Convergence and divergence of series and sequences |
| 11B05 | Density, gaps, topology |
| 11B50 | Sequences (mod \(m\)) |
| 11K31 | Special sequences |
Keywords:
statistically convergent sequence; statistical limit point; asymptotic density; distribution function of a sequenceReferences:
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