zbMATH — the first resource for mathematics

Statistical limit points. (English) Zbl 0776.40001
Summary: Following the concept of a statistically convergent sequence \(x\), we define a statistical limit point of \(x\) as a number \(\lambda\) that is the limit of a subsequence \(\{x_{k(j)}\}\) of \(x\) such that the set \(\{k(j):j\in{\mathbf N}\}\) does not have density zero. Similarly, a statistical cluster point of \(x\) is a number \(\gamma\) such that for every \(\varepsilon>0\) the set \(\{k\in\mathbb{N}:| x_ k- \gamma|<\varepsilon\}\) does not have density zero. These concepts, which are not equivalent, are compared to the usual concept of limit point of a sequence. Statistical analogues of limit point results are obtained. For example, if \(x\) is a bounded sequence then \(x\) has a statistical cluster point but not necessarily a statistical limit point. Also, if the set \(M:=\{k\in\mathbb{N}:x_ k>x_{k+1}\}\) has density one and \(x\) is bounded on \(M\), then \(x\) is statistically convergent.

40A05 Convergence and divergence of series and sequences
26A03 Foundations: limits and generalizations, elementary topology of the line
11B05 Density, gaps, topology
Full Text: DOI
[1] R. Creighton Buck, The measure theoretic approach to density, Amer. J. Math. 68 (1946), 560 – 580. · Zbl 0061.07503
[2] J. S. Connor, The statistical and strong \?-CesĂ ro convergence of sequences, Analysis 8 (1988), no. 1-2, 47 – 63. · Zbl 0653.40001
[3] Jeff Connor, \?-type summability methods, Cauchy criteria, \?-sets and statistical convergence, Proc. Amer. Math. Soc. 115 (1992), no. 2, 319 – 327. · Zbl 0765.40002
[4] H. Fast, Sur la convergence statistique, Colloquium Math. 2 (1951), 241 – 244 (1952) (French). · Zbl 0044.33605
[5] J. A. Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301 – 313. · Zbl 0588.40001
[6] J. A. Fridy and H. I. Miller, A matrix characterization of statistical convergence, Analysis 11 (1991), no. 1, 59 – 66. · Zbl 0727.40001
[7] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. · Zbl 0281.10001
[8] Ivan Niven and Herbert S. Zuckerman, An introduction to the theory of numbers, 4th ed., John Wiley & Sons, New York-Chichester-Brisbane, 1980. · Zbl 0431.10001
[9] I. J. Schoenberg, The integrability of certain functions and related summability methods., Amer. Math. Monthly 66 (1959), 361 – 375. · Zbl 0089.04002
[10] A. Zygmund, Trigonometric series, 2nd ed., vol. II, Cambridge Univ. Press, London and New York, 1979. · JFM 58.0296.09
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.