Let \(X\) be a Banach space ordered by a closed convex cone \(X_+\) with nonempty interior such that the norm is monotonic. Denote by \({\mathfrak M}\) the set of all sequences of monotonic linear operators. For \(\eta \in \text{int}(X_+)\) the interval \(\langle -\eta,\eta \rangle\) induces an equivalent norm on \(X\). This norm leads to two metrizable topologies (strong and weak) on various subspaces of \({\mathfrak M}\). The authors present several results about the asymptotic behavior of infinite products of generic elements of these subspaces. In addition to a weak ergodic theorem the authors also obtain convergence to a 1-dimensional operator \(f \otimes \eta\) where \(f\) is a continuous linear functional and \(\eta\) is a common fixed point.