 Ergodicity, finally, implies no decay of correlation at all. The ergodicity condition only says that the average of the correlations (and this time without an absolute value) of all past events with a  future event is zero. But this is compatible with there being strong correlations between every instant in the past and the future, provided that positive and negative correlations average out. So in ergodic systems the past does not become irrelevant. For this reason ergodic system are not random at all (in the sense of random introduced above).