  We study the convergence in rough path topology of a certain class of
discrete processes, the hidden Markov walks, to a Brownian motion with an area
anomaly. This area anomaly, which is a new object, keeps track of the
time-correlation of the discrete models and brings into light the question of
embeddings of discrete processes into continuous time. We also identify an
underlying combinatorial structure in the hidden Markov walks, which turns out
to be a generalization of the occupation time from the classical ergodic
theorem in the spirit of rough paths.
