Distinguishing Between Contagion, Heterogeneity and Randomness in Stochastic Models

It is commonly believed that the nature of the generating structure of a stochastic process can be determined by examining the distributional form of the stochastic process in time. For example, it is believed that correlations between time intervals can be used to distinguish between contagion (after-effect) and heterogeneity (stratification). These beliefs are incorrect, and no amount of statistical information internal to the sampled process, is sufficient for such a distinction.