A new equivalence notion between non-stationary subdivision schemes, termed
asymptotical similarity, which is weaker than asymptotical equivalence, is
introduced and studied. It is known that asymptotical equivalence between a
non-stationary subdivision scheme and a convergent stationary scheme guarantees
the convergence of the non-stationary scheme. We show that for non-stationary
schemes reproducing constants, the condition of asymptotical equivalence can be
relaxed to asymptotical similarity. This result applies to a wide class of
non-stationary schemes of importance in theory and applications
