For a sequence-to-sequence transformation A , let R m A x = ∑ n ≥ m | ( A x ) n | and μ m A x = sup n ≥ m | ( A x ) n | . The purpose of this paper is to study the relationship between the asymptotic equivalence of two sequences ( lim n x n / y n = 1 ) and the variations of asymptotic equivalence based on the ratios R m A x / R m A y and μ m A x / μ m A y .