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 [x] and [y] are S σ,8-asymptotically equivalent of multiple L provided that for every ε > 0 $$\underset{r}{lim}\frac{1}{h_r}\left\{k\in I_r:\left|\frac{x_{{\sigma }^k\left(m\right)}}{y_{{\sigma }^k\left(m\right)}}-L\right|⩾\in \right\}=0$$ uniformly in m = 1, 2, 3, ..., (denoted by x $$\stackrel{S_{\sigma ,\theta }}{\sim }$$ y) simply S σ,8-asymptotically equivalent, if L = 1. Using this definition we shall prove S σ,8-asymptotically equivalent...