  This paper investigates the asymptotic behaviour of solutions to certain
infinite systems of coupled recurrence relations. In particular, we obtain a
characterisation of those initial values which lead to a convergent solution,
and for initial values satisfying a slightly stronger condition we obtain an
optimal estimate on the rate of convergence. By establishing a connection with
a related problem in continuous time, we are able to use this optimal estimate
to improve the rate of convergence in the continuous setting obtained by the
authors in a previous paper. We illustrate the power of the general approach by
using it to study several concrete examples, both in continuous and in discrete
time.
