Strongly statistical convergence

Abstract

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.