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 ∆ m lacunary statistical(defined in [2]) equivalent of multiple L provided that for every > 0, lim r 1 hr { the number of k ∈ Ir : ∣∣∣∣∆mxk ∆yk − L ∣∣∣∣ ≥ } = 0 (denoted by x S θ (∆ ) ∼ y), and simply ∆m-lacunary asymptotically statistical equivalent if L = 1. In section 3 are given some properties of ∆m-statistical asymptotically equivalent sequences and ∆m-Cesaro asymptotically equivalent sequences. In Theorems 3.1, 3.3, 3.4, 3.6, 3.7 are given inclusion cases of those classes and in Theorems 3.5, 3.8 and 3.9 are given equivalent conditions of those classes. In section 4 are given ∆m-Cesaro Orlicz asymptotically equivalent sequences and their relationship with other classes. In Theorems 4.3, 4.4, 4.5 and 4.6 are given inclusion cases of this class of sequences and classes defined in section 3. By Theorems 4.7, 4.8 and 4.9 are given equivalent conditions for those classes. AMS Mathematics Subject Classification (2010): 40A05, 40C05, 46A45.