Stationary Stochastic Processes are Mixing of Ergodic Ones: Contingency

Abstract

Stationarity of a stochastic process seems connected to the idea of constancy. But ergodicity is needed for the property that almost surely the observation of a trajectory from time -∞ to 0 makes possible the identification of the law of the whole process, including the future. When the stationary process is a Markov chain with a finite number of states it is well known that the set of states divides into ergodic classes. Decomposition of more general stationary processes in ergodic classes goes back to von Neumann. This result has been improved and/or rediscovered several times, and it received a lot of different proofs. Its philosophical interpretation as the concept of contingency does not seem given in the literature. After some preliminaries we will survey a part of the most basic results