  The concept of effective complexity of an object as the minimal description
length of its regularities has been initiated by Gell-Mann and Lloyd. The
regularities are modeled by means of ensembles, that is probability
distributions on finite binary strings. In our previous paper we propose a
definition of effective complexity in precise terms of algorithmic information
theory. Here we investigate the effective complexity of binary strings
generated by stationary, in general not computable, processes. We show that
under not too strong conditions long typical process realizations are
effectively simple. Our results become most transparent in the context of
coarse effective complexity which is a modification of the original notion of
effective complexity that uses less parameters in its definition. A similar
modification of the related concept of sophistication has been suggested by
Antunes and Fortnow.
