  Andresen and Spokoiny's (2013) ``critical dimension in semiparametric
estimation`` provide a technique for the finite sample analysis of profile
M-estimators. This paper uses very similar ideas to derive two convergence
results for the alternating procedure to approximate the maximizer of random
functionals such as the realized log likelihood in MLE estimation. We manage to
show that the sequence attains the same deviation properties as shown for the
profile M-estimator in Andresen and Spokoiny (2013), i.e. a finite sample Wilks
and Fisher theorem. Further under slightly stronger smoothness constraints on
the random functional we can show nearly linear convergence to the global
maximizer if the starting point for the procedure is well chosen.
