Probabilistic Formulation: Birkhoff–Khinchin Theorem
Birkhoff–Khinchin theorem. Let ƒ be measurable, E(|ƒ|) < ∞, and T be a measure-preserving map. Then with probability 1:
where is the conditional expectation given the σ-algebra of invariant sets of T.
Corollary (Pointwise Ergodic Theorem): In particular, if T is also ergodic, then is the trivial σ-algebra, and thus with probability 1:
Read more about this topic: Ergodic Theory
Famous quotes containing the word theorem:
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)