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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

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

How this entity was disambiguated

This entity first appeared as the object of triple T6293667 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: ergodic theorem
Context triple: [law of large numbers, relatesTo, ergodic theorem]

A. Kakutani equivalence in ergodic theory
Kakutani equivalence in ergodic theory is a notion of equivalence between measure-preserving dynamical systems based on the isomorphism of their induced transformations on subsets of positive measure.
B. Kakutani’s random ergodic theorem
Kakutani’s random ergodic theorem is a fundamental result in ergodic theory that extends classical ergodic theorems to sequences of randomly chosen measure-preserving transformations.
C. Lectures on Ergodic Theory
"Lectures on Ergodic Theory" is a classic mathematical monograph that systematically develops the foundations and key results of ergodic theory within dynamical systems.
D. Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
E. Poincaré recurrence theorem
The Poincaré recurrence theorem is a fundamental result in dynamical systems and ergodic theory stating that certain systems will, after a sufficiently long but finite time, return arbitrarily close to their initial state.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: ergodic theorem
Target entity description: 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.

A. Kakutani equivalence in ergodic theory
Kakutani equivalence in ergodic theory is a notion of equivalence between measure-preserving dynamical systems based on the isomorphism of their induced transformations on subsets of positive measure.
B. Kakutani’s random ergodic theorem
Kakutani’s random ergodic theorem is a fundamental result in ergodic theory that extends classical ergodic theorems to sequences of randomly chosen measure-preserving transformations.
C. Lectures on Ergodic Theory
"Lectures on Ergodic Theory" is a classic mathematical monograph that systematically develops the foundations and key results of ergodic theory within dynamical systems.
D. Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
E. Poincaré recurrence theorem
The Poincaré recurrence theorem is a fundamental result in dynamical systems and ergodic theory stating that certain systems will, after a sufficiently long but finite time, return arbitrarily close to their initial state.
F. None of above. chosen

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 ⓘ

How these facts were elicited

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 ⓘ