Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Statistical limit points


Author: J. A. Fridy
Journal: Proc. Amer. Math. Soc. 118 (1993), 1187-1192
MSC: Primary 40C99
DOI: https://doi.org/10.1090/S0002-9939-1993-1181163-6
MathSciNet review: 1181163
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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 \mathbb{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}:\vert{x_k} - \gamma \vert < \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.


References [Enhancements On Off] (What's this?)

  • [1] R. C. Buck, The measure theoretic approach to density, Amer. J. Math. 68 (1946), 560-580. MR 0018196 (8:255f)
  • [2] J. S. Connor, The statistical and strong $ p$-Cesáro convergence of sequences, Analysis 8 (1988), 47-63. MR 954458 (89k:40013)
  • [3] -, $ R$-type summability methods, Cauchy criteria, $ p$-sets, and statistical convergence, Proc. Amer. Math. Soc. 115 (1992), 319-327. MR 1095221 (92i:40005)
  • [4] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244. MR 0048548 (14:29c)
  • [5] J. A. Fridy, On statistical convergence, Analysis 5 (1985), 301-313. MR 816582 (87b:40001)
  • [6] J. A. Fridy and H. I. Miller, A matrix characterization of statistical convergence, Analysis 11 (1991), 59-66. MR 1113068 (92e:40001)
  • [7] L. Kupers and M. Niederreiter, Uniform distribution of sequences, Wiley, New York, 1974. MR 0419394 (54:7415)
  • [8] I. Niven and H. S. Zuckerman, An introduction to the theorem of numbers, 4th ed., Wiley, New York, 1980. MR 572268 (81g:10001)
  • [9] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361-375. MR 0104946 (21:3696)
  • [10] A. Zygmund, Trigonometric series, 2nd ed., vol. II, Cambridge Univ. Press, London and New York, 1979.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 40C99

Retrieve articles in all journals with MSC: 40C99


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1181163-6
Keywords: Natural density, statistically convergent sequence
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society