ergodic theorem

E582382

mathematical theorem result in ergodic theory

The ergodic theorem is a fundamental result in dynamical systems and probability theory that links long-term time averages of a system’s evolution to ensemble or space averages, underpinning the statistical behavior of many physical and stochastic processes.

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All labels observed (6)

Label	Occurrences
Birkhoff ergodic theorem	2
Birkhoff’s ergodic theorem	1
Birkhoff’s pointwise ergodic theorem	1
ergodic theorem canonical	1
von Neumann mean ergodic theorem	1
von Neumann’s mean ergodic theorem	1

Statements (51)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in ergodic theory ⓘ
appliesTo	Markov chains with unique stationary distribution ⓘ ergodic transformations ⓘ measure-preserving dynamical systems ⓘ stationary ergodic processes ⓘ
field	dynamical systems ⓘ ergodic theory ⓘ probability theory ⓘ statistical mechanics ⓘ
generalizationOf	law of large numbers ⓘ
hasConsequence	almost sure convergence of time averages ⓘ existence of typical trajectories ⓘ law of large numbers for dynamical systems ⓘ
hasForm	Birkhoff ergodic theorem NERFINISHED ⓘ ergodic theorem for Markov chains ⓘ ergodic theorem for flows ⓘ ergodic theorem for group actions ⓘ ergodic theorem for stationary processes ⓘ mean ergodic theorem NERFINISHED ⓘ multiparameter ergodic theorem ⓘ pointwise ergodic theorem NERFINISHED ⓘ subadditive ergodic theorem NERFINISHED ⓘ von Neumann mean ergodic theorem NERFINISHED ⓘ
historicalDevelopment	formulated in modern form in the 20th century ⓘ
implies	equivalence of time and ensemble averages under ergodicity ⓘ
notableContributor	George David Birkhoff NERFINISHED ⓘ John von Neumann NERFINISHED ⓘ
relatedTo	Kolmogorov–Sinai entropy NERFINISHED ⓘ Poincaré recurrence theorem NERFINISHED ⓘ mixing ⓘ strong mixing ⓘ weak mixing ⓘ
relatesConcept	ensemble average ⓘ ergodic measure ⓘ invariant measure ⓘ long-term statistical behavior ⓘ measure-preserving transformation ⓘ space average ⓘ stationary stochastic process ⓘ time average ⓘ
states	for an ergodic measure-preserving transformation, time averages converge almost surely to space averages ⓘ
usedIn	Markov chain Monte Carlo NERFINISHED ⓘ Monte Carlo methods NERFINISHED ⓘ chaos theory ⓘ information theory ⓘ probabilistic number theory ⓘ random dynamical systems ⓘ signal processing ⓘ statistical mechanics ⓘ thermodynamics ⓘ

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

law of large numbers → relatesTo → ergodic theorem ⓘ

Garrett Birkhoff → notableConcept → ergodic theorem ⓘ

this entity surface form: Birkhoff’s ergodic theorem

Kakutani’s random ergodic theorem → generalizes → ergodic theorem ⓘ

this entity surface form: Birkhoff’s pointwise ergodic theorem

Kakutani’s random ergodic theorem → generalizes → ergodic theorem ⓘ

this entity surface form: von Neumann’s mean ergodic theorem

Lectures on Ergodic Theory → subject → ergodic theorem ⓘ

this entity surface form: Birkhoff ergodic theorem

Lectures on Ergodic Theory → subject → ergodic theorem ⓘ

this entity surface form: von Neumann mean ergodic theorem

Khinchin–Lévy constant → relatedConcept → ergodic theorem ⓘ

this entity surface form: Birkhoff ergodic theorem