Ergodic Properties

The central notion of ergodic theory is a measurable operation on points of a probability space. This operation corresponds to letting the random system evolve freely until the next experiment. This chapter describes what happens to the random variables on average when the measurable operation is repeated. It shows that a restricted frequency interpretation of probability holds true in some important cases. The respective result is called the Birkhoff ergodic theorem. The chapter derives several practical criteria to inspect the algebraic properties of the probability measure and states some theorems showing how taking products or functions of stochastic processes affects their ergodicity and mixing. Ergodic decomposition theorems allow the representation of any stationary measure as an integral over the space of stationary ergodic measures.