On $\lambda^2$-asymptotically double statistical equivalent sequences

Authors

1 Adiyaman University, Science and Art Faculty, Department of Mathematics, 02040, Adiyaman, Turkey

2 Gaziantep University, Science and Art Faculty, Department of Mathematics, 27200, Gaziantep,Turkey

Abstract

This paper presents the following new definition which is a natural combination of the definition for asymptotically double equivalent, double statistically limit and double $\lambda^2$-sequences. The double sequence $\lambda^2 = (\lambda_{m,n})$ of positive real numbers tending to infinity such that
$$\lambda_{m+1,n}\leq\lambda_{m,n} + 1,  \lambda_{m,n+1}\leq\lambda{m,n} + 1,$$
$$\lambda_{m,n} -\lambda_{m+1,n }\leq\lambda_{m,n+1}\lambda_{m+1,n+1},  \lambda_{1,1} = 1,$$
and
$$I_{m,n}=\{(k,l) : m -\lambda_{m,n }+ 1 \leq k \leq m,   n -\lambda_{m,n} + 1 \leq l \leq n.$$
For double $\lambda^2$-sequence; the two non-negative sequences $x = (x_{k,l})$ and $y = (y_{k,l})$ are said to be
$\lambda^2$-asymptotically double statistical equivalent of multiple $L$ provided that for every $\varepsilon> 0$
$$P - \lim_{m,n}\frac{1}{\lambda_{m,n}}|\{(k,l)\in I_{m,n}:|\frac{x_{k,l}}{y_{k,l}}-L\geq\varepsilon\}|=0$$
(denoted by $x\sim^{S_{\lambda^2}^L } y$) and simply $\lambda^2$-asymptotically double statistical equivalent if $L = 1$.

Keywords