On ℐ-asymptotically lacunary statistical equivalent sequences

Ekrem Savaş

doi:10.1186/1687-1847-2013-111

Abstract

This paper presents the following definition, which is a natural combination of the definitions for asymptotically equivalent, ℐ-statistically limit and ℐ-lacunary statistical convergence. Let θ be a lacunary sequence; the two nonnegative sequences x=( x k ) and y=( y k ) are said to be ℐ-asymptotically lacunary statistical equivalent of multiple L provided that for every ϵ>0, and δ>0, { r ∈ N : 1 h r | { k ∈ I r : | x k y k − L | ≥ ε } | ≥ δ } ∈I (denoted by x ∼ S θ L ( I ) y) and simply ℐ-asymptotically lacunary statistical equivalent if L=1.MSC:40A99, 40A05.

Highlights

In, Marouf [ ] presented definitions for asymptotically equivalent sequences and asymptotic regular matrices
In [ ], asymptotically lacunary statistical equivalent, which is a natural combination of the definitions for asymptotically equivalent, statistical convergence and lacunary sequences
We introduce two new notions I-asymptotically lacunary statistical equivalent of multiple L and strong I-asymptotically lacunary equivalent of multiple L

Summary

Introduction

In , Marouf [ ] presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. (Marouf [ ]) Two nonnegative sequences x = (xk) and y = (yk) are said to be asymptotically equivalent if lim xk = k yk (denoted by x ∼ y). (Patterson [ ]) Two nonnegative sequences x = (xk) and y = (yk) are said to be asymptotically statistical equivalent of multiple L provided that for every > ,

Results

Conclusion