@article{Kaya_Aral_2020, title={Strongly statistical convergence}, volume={72}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/2368}, abstractNote={<p>UDC 519.21</p> <p>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. <br>A sequence $\left(x_{n}\right)$ is called strongly convergent to $L$ if $\displaystyle{\lim
olimits_{n\to\infty} \sum
olimits_{k=1}^{\infty}a_{nk}\left|x_{k}-L\right|=0}$ in the ordinary sense. <br>In the definition of $A$-strongly statistical limit, we use the statistical limit instead of the ordinary limit via a convenient density. <br>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. <br>We introduce the notion of dense positivity for nonnegative sequences. <br>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. <br>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 $\displaystyle{r_{n}=\sum
olimits_{k=1}^{\infty}a_{nk }.$ <br>Furthermore, necessary conditions for the regularity, linearity and multiplicativity of $A$-strongly statistical limit are established.</p>}, number={2}, journal={Ukrainsâ€™kyi Matematychnyi Zhurnal}, author={KayaU. and Aral N. D.}, year={2020}, month={Feb.}, pages={221-231} }