Ergodic Theory

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Ergodic Theory is about random processes in discrete or continuous time. Such a process is called ergodic if all averages over time are the same with probability one, and equal to the expectation value of the quantity considered. A more mathematical definition is: the probability distribution of the process is extremal among the shift invariant ones. The term was coined by Boltzmann in the 19-th century to designate a hypothesis which he made about statistical mechanical systems in order to justify the use of equilibrium probability densities. The first ergodic theorems were proved by von Neumann (in the L^2 sense) and Birkhoff (with probability 1).