Strongly statistical convergence

  • U. Kaya Bitlis Eren Univ., Turkey
  • N. D. Aral Bitlis Eren Univ., Turkey
Keywords: convergence


UDC 519.21

We introduce $A$-strongly statistical convergence for sequences of complex numbers, where $A=\left(a_{nk}\right)_{n,k\in \mathbb{N}}$ is an infinite matrix with nonnegative entries.
A sequence $\left(x_{n}\right)$ is called strongly convergent to $L$ if $\displaystyle{\lim\nolimits_{n\to\infty} \sum\nolimits_{k=1}^{\infty}a_{nk}\left|x_{k}-L\right|=0}$ in the ordinary sense.
In the definition of $A$-strongly statistical limit, we use the statistical limit instead of the ordinary limit via a convenient density.
We study some densities and show that the $\left(a_{nk}\right)$-strongly statistical limit is a $\left(a_{m_{n}k}\right)$-strong limit, where the density of the set $\left\{m_{n}\in\mathbb{N}\colon n\in\mathbb{N}\right\}$ is equal to 1.
We introduce the notion of dense positivity for nonnegative sequences.
A nonnegative sequence $\left(r_{n}\right)$ is dense positive provided the limit superior of a subsequence $\left(r_{m_{n}}\right)$ is positive for all $\left(m_{n}\right)$ with density equal to 1.
We show that the dense positivity of $\left(r_{n}\right)$ is a necessary and sufficient condition for the uniqueness of $A$-strongly statistical limit, where $A=\left(a_{nk}\right)$ and $\displaystyle{r_{n}=\sum\nolimits_{k=1}^{\infty}a_{nk}}.$
Furthermore, necessary conditions for the regularity, linearity and multiplicativity of $A$-strongly statistical limit are established.


Belen, C.; Mohiuddine, S. A. Generalized weighted statistical convergence and application. Appl. Math. Comput. 219 (2013), no. 18, 9821--9826. doi: 10.1016/j.amc.2013.03.115

Connor, Jeff. On strong matrix summability with respect to a modulus and statistical convergence. Canad. Math. Bull. 32 (1989), no. 2, 194--198. doi: 10.4153/CMB-1989-029-3

Edely, Osama H. H.; Mursaleen, M.; Khan, Asif. Approximation for periodic functions via weighted statistical convergence. Appl. Math. Comput. 219 (2013), no. 15, 8231--8236. doi: 10.1016/j.amc.2013.02.024

Fast, H. Sur la convergence statistique. (French) Colloq. Math. 2 (1951), 241--244 (1952). doi: 10.4064/cm-2-3-4-241-244

Freedman, A. R.; Sember, J. J. Densities and summability. Pacific J. Math. 95 (1981), no. 2, 293--305. MR0632187

Fridy, J. A. On statistical convergence. Analysis 5 (1985), no. 4, 301--313. doi: 10.1524/anly.1985.5.4.301

Ghosal, Sanjoy. Generalized weighted random convergence in probability. Appl. Math. Comput. 249 (2014), 502--509. doi: 10.1016/j.amc.2014.10.056

Hamilton, H. J.; Hill, J. D. On Strong Summability. Amer. J. Math. 60 (1938), no. 3, 588--594. doi: 10.2307/2371600

Karakaya, V.; Chishti, T. A. Weighted statistical convergence. Iran. J. Sci. Technol. Trans. A Sci. 33 (2009), no. 3, 219--223 (2010). MR2848371

Maddox, I. J. Spaces of strongly summable sequences. Quart. J. Math. Oxford Ser. (2) 18 (1967), 345--355. doi: 10.1093/qmath/18.1.345

I. J. Maddox, Elements of Functional Analysis, Cambridge, UK: Cambridge Univ. Press (1970).

Mursaleen, Mohammad; Karakaya, Vatan; Ertürk, Müzeyyen; Gürsoy, Faik. Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl. Math. Comput. 218 (2012), no. 18, 9132--9137. doi: 10.1016/j.amc.2012.02.068

Šalát, T. On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), no. 2, 139--150. MR0587239

Steinhaus, H. Quality control by sampling (a plea for Bayes' rule). Colloq. Math. 2 (1951), 98--108. doi: 10.4064/cm-2-2-98-108

Włodarski, L. On some strong continuous summability methods. Proc. London Math. Soc. (3) 13 1963 273--289. doi: 10.1112/plms/s3-13.1.273

How to Cite
Kaya, U., and N. D. Aral. “Strongly Statistical Convergence”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 2, Feb. 2020, pp. 221-3,
Research articles