On asymptotically D m lacunary statistical equivalent sequences N.L. Braha Department of Mathematics and Computer Sciences, Avenue Mother Teresa, 10000, Prishtine Kosova article info Keywords: Difference sequence Lacunary sequence Statistical convergence Asymptotically equivalent Orlicz function abstract This paper presents the following definition which is a natural combination of the defini- tion for asymptotically equivalent, statistically limit and lacunary sequences. Let h be a lacunary sequence; the two nonnegative sequences x ¼ðx k Þ and y ¼ðy k Þ are said to be asymptotically D m lacunary statistical (defined in [2]) equivalent of multiple L provided that for every  > 0, lim r 1 h r the number of k 2 I r : D m x k D m y k  L         P    ¼ 0 (denoted by x  S L h ðD m Þ y), and simply D m -lacunary asymptotically statistical equivalent if L ¼ 1. Also are given some properties of D m -statistical asymptotically equivalent sequences and D m -Cesaro asymptotically equivalent sequences and inclusion cases of those classes, more over, equivalent conditions of those classes. In last section are given D m -Cesaro Orlicz asymptotically equivalent sequences and their relationship with other classes, such as: inclusion cases of this class of sequences and classes defined in Section 3 and equivalent conditions for those classes. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction In 1993, Marouf presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. In 2003, Patterson extend these concepts by presenting an asymptotically statistical equivalent analog of these definitions and nat- ural regularity conditions for nonnegative summability matrices. In [11], Patterson and Savas extend these concepts to asymptotically lacunary statistical equivalent sequences. In this paper we will introduce the asymptotically D m lacunary sta- tistical equivalent sequences, where concept of D m lacunary statistical sequences is introduced in [2]. This concept is exten- sion of the previous concepts of asymptotically sequences. In addition to these definition, natural inclusion theorems shall also be presented. Also we will present Cesaro Orlicz asymptotically statistical equivalent and lacunary Cesaro Orlicz asymp- totically statistical equivalent sequences. 2. Definitions and notations Definition 2.1 (Marouf [9]). Two nonnegative sequences ðx k Þ and ðy k Þ are said to be asymptotically equivalent if 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.06.016 E-mail address: nbraha@yahoo.com Applied Mathematics and Computation 219 (2012) 280–288 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc