  We prove that the fixed point iteration of arbitrary positive concave
mappings with nonempty fixed point set converges geometrically for any starting
point. We also show that positivity is crucial for this result to hold, and the
concept of (nonlinear) spectral radius of asymptotic mappings provides us with
information about the convergence factor. As a practical implication of the
results shown here, we rigorously explain why some power control and load
estimation algorithms in wireless networks, which are particular instances of
the fixed point iteration, have shown geometric convergence in simulations.
These algorithms have been typically derived by considering fixed point
iterations of the general class of standard interference mappings, so the
possibility of sublinear convergence rate could not be ruled out in previous
studies, except in special cases that are often more restrictive than those
considered here.
