Motivated by the works of Knuth, we introduce several asymptotic notions, such as Oab((yn)), Ωab((yn)) and Θab((yn)) for arbitrary sequential convergences a and b over a vector lattice X. The central idea is to compare the convergence rate of an a-convergent sequence with that of a b-null sequence. After exposing the relations of these asymptotic notations to lattice operations, we examine how they can be used with several iterative procedures, such as the modified Newton method. We further utilize these asymptotic notations to study the rates of asymptotically abelian sequences. Several examples and detailed discussions of various cases are given.