On asymptotically Δm lacunary statistical equivalent sequences

This paper presents the following definition which is a natural combination of the definition for asymptotically equivalent, statistically limit and lacunary sequences. Let θ be a lacunary sequence; the two nonnegative sequences x=(xk) and y=(yk) are said to be asymptotically Δm lacunary statistical (defined in [2]) equivalent of multiple L provided that for every ∊>0,limr1hrthe number ofk∈Ir:ΔmxkΔmyk-L⩾∊=0(denoted by x∼SθL(Δm)y), and simply Δm-lacunary asymptotically statistical equivalent if L=1. Also are given some properties of Δm-statistical asymptotically equivalent sequences and Δm-Cesaro asymptotically equivalent sequences and inclusion cases of those classes, more over, equivalent conditions of those classes. In last section are given Δm-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.