One of the distinguishing features of the present scientific method is the notion of reproducibility. In situations subject to random perturbations reproducibility allows averaging over many individual experiments in order to remove randomness and arrive at near certain statements.
When one assumes that events further and further in the past have less and less influence on the present, the probabilistic paradigm is well understood and is successful in many scientific and technological applications. However, some important applications lead to stochastic processes whose present outcomes are significantly influenced by events in the remote past.
Salient examples occur in the theory of random walks, where there is a classical dichotomy between recurrent and transient behaviour. We present a very simple example of a random walk with infinite memory which is neither known to be transient nor recurrent. Then, using a reinforcement mechanism due to Polya, we explain the nature of a particular infinite memory process in terms of spontaneous emergence of opinions. Finally we mention some recent results towards understanding the recurrence-transience dichotomy for reinforced random walks, and indicate an application to coding used in optical CD technology.