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

中文翻译：

---

统计收敛性强

我们为复数序列引入了 A-强统计收敛的概念，其中 A = (ank)n,k∈ℕ 是具有非负项的无限矩阵。如果 $$ \underset{n\to \infty }{\lim }{\sum}_{k=1}^{\infty }{a}_{nk}\ left|{x}_k\right.-L\left|=0\right. 普通意义上的 $$。在A-强统计极限的定义中，我们用统计极限代替了密度方便的普通极限。我们研究了一些密度并表明 (ank)-强统计极限是一个 ( $$ {a}_{m_nk} $$ )-强极限，其中集合 {mn ∈ ℕ : n ∈ ℕ} 的密度是等于 1。我们为非负序列引入了密集正性的概念。一个非负序列 (rn) 是密集正的，前提是子序列 ( $$ {r}_{m_n} $$ ) 对于所有密度等于 1 的 (mn) 都是正的。我们证明了 ( rn) 是 A-强统计极限唯一性的充要条件，其中 A = (ank) 且 rn = $$ {\sum}_{k=1}^{\infty {a}_{nk } $$ 。进一步建立了A-强统计极限的正则性、线性和可乘性的必要条件。