On Asymptotically Lacunary Statistical Equivalent Sequences

F. Patterson, E. Savas

Abstract

This paper presents the following definition which is a natural combination of the definition for asymptotically equivalent, statistically limit and lacunary sequences. Let µ be a lacunary sequence; the two nonnegative sequences [x] and [y] are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every $ \epsilon > 0 $

$\lim_r(1/h_r)|{k\in I_r : |(x_k /y_k)-L|\geq \epsilon }|=0 $

(denoted by $x \sim^{S^L_\theta} y$ and simply asymptotically lacunary statistical equivalent if L= 1. In addition, we shall also present asymptotically equivalent analogs of Fridy's and Orhan's theorems in [3].