Asymptotically Lacunary Statistically Equivalent Sequences of Interval Numbers

Abstract

In this article we present the following definition of asymptotic equivalence
which is natural combination of the definition for asymptotically equivalent and lacunary statistical convergence of interval numbers. Let $\theta=\left(
k_{r}\right) $ be a lacunary sequence, then the two sequnces $\overline{x}=\left(\overline{x}_{k}\right)$ and $0\notin$ $\overline{y}=\left(\overline{y}_{k}\right) $ of interval numbers are said to be asymptotically lacunary statistically equivalent to multiple $\overline{1}=\left[1,1\right]$ provided that for every $\varepsilon>0$
\[
\lim_{r}\frac{1}{h_{r}}\left\vert \left\{ k\in I_{r}:d\left( \frac
{\overline{x}_{k}}{\overline{y}_{k}},\overline{1}\right) \geq\varepsilon
\right\} \right\vert =0.
\]