Σ-Asymptotically Lacunary Statistical Equivalent Sequences

Abstract

This paper presents the following definitions which is a natural combination of the definition for asymptotically equivalent, statistically limit, lacunary sequences, and σ-convergence. Let θ be a lacunary sequence; Two nonnegative sequences [χ] and [y] are Sσ,θ-asymptotically equivalent of multiple L provided that for every ε > 0 limr 1/hr {k ε Ir: xσκ(m)/yσk(m) - L ≥ ε} = 0 uniformly in m = 1, 2, 3,..., (denoted by x ∼ Sσ,θy) simply Sσ,θ -asymptotically equivalent, if L = 1. Using this definition we shall prove Sσ,θ-asymptotically equivalent analogues of Fridy and Orhan's theorems in [5] and analogues results of Das and Patel in [1] shall also be presented. © Versita Warsaw and Springer-Verlag Berlin Heidelberg 2006.