  In this paper, we study the asymptotic behavior of lengths of $\tor$ modules
of homologies of complexes under the iterations of the Frobenius functor in
positive characteristic. We first give upper bounds to this type of length
functions in lower dimensional cases and then construct a counterexample to the
general situation. The motivation of studying such length functions arose
initially from an asymptotic length criterion given in [D4] which is a
sufficient condition to a special case of nonnegativity of $\chi_\infty$. We
also provide an example to show that this sufficient condition does not hold in
general.
