  In this paper we consider unbounded solutions of perturbed convolution
Volterra summation equations. The equations studied are asymptotically
sublinear, in the sense that the state--dependence in the summation is of
smaller than linear order for large absolute values of the state. When the
perturbation term is unbounded, it is elementary to show that solutions are
also. The main results of the paper are mostly of the following form: the
solution has an additional unboundedness property $U$ if and only if the
perturbation has property $U$. Examples of property $U$ include monotone
growth, monotone growth with fluctuation, fluctuation on $\mathbb{R}$ without
growth, existence of time averages. We also study the connection between the
times at which the perturbation and solution reach their running maximum, and
the connection between the size of signed and unsigned running maxima of the
solution and forcing term.
