Strongly Statistical Convergence

We introduce a concept of A-strongly statistical convergence for sequences of complex numbers, where A = (a_nk)_n,k∈ℕ is an infinite matrix with nonnegative entries. A sequence (x_n) 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 (a_nk)-strongly statistical limit is an (${a}_{m_nk}$)-strong limit, where the density of the set {m_n ∈ ℕ : n ∈ ℕ} is equal to 1. We introduce the notion of dense positivity for nonnegative sequences. A nonnegative sequence (r_n) is dense positive provided the limit superior of a subsequence (${r}_{m_n}$) is positive for all (m_n) with density equal to 1. We show that the dense positivity of (r_n) is a necessary and sufficient condition for the uniqueness of A-strongly statistical limit, where A = (a_nk) and r_n = ${\sum}_{k=1}^{\infty }{a}_{nk}$. Furthermore, necessary conditions for the regularity, linearity and multiplicativity of the A-strongly statistical limit are established.