Kac's lemma
E620664
Kac's lemma is a result in ergodic theory that relates the expected return time to a set in a measure-preserving dynamical system to the inverse of the measure of that set.
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
result in ergodic theory
ⓘ
theorem ⓘ |
| appearsIn | Mark Kac's works on probability and statistical mechanics ⓘ |
| appliesTo |
ergodic measure-preserving transformations
ⓘ
measure-preserving dynamical systems ⓘ |
| assumes |
measurable set of positive measure
ⓘ
measure-preserving transformation ⓘ probability space ⓘ |
| category |
theorems in dynamical systems
ⓘ
theorems in ergodic theory ⓘ theorems in probability theory ⓘ |
| conclusion | E(τ_A | x in A) = 1/μ(A) where τ_A is the first return time to A ⓘ |
| consequence |
recurrence properties can be quantified via measure
ⓘ
the expected number of visits to a set is proportional to its measure ⓘ |
| field |
dynamical systems
ⓘ
ergodic theory ⓘ probability theory ⓘ |
| formalSetting | (X, Σ, μ, T) with μ a probability measure and T measure-preserving ⓘ |
| hasConcept |
ergodic transformation
ⓘ
first return time ⓘ invariant measure ⓘ recurrence time ⓘ |
| holdsIn |
discrete-time dynamical systems
ⓘ
measure-preserving transformations on probability spaces ⓘ |
| implies |
the average return time to a set is inversely proportional to its measure
ⓘ
the mean recurrence time to a set is finite if the set has positive measure ⓘ |
| influenced |
applications of ergodic theory to Markov chains
ⓘ
study of return-time statistics in dynamical systems ⓘ |
| isSpecialCaseOf | results on return times in stationary processes ⓘ |
| mathematicalDomain |
measure theory
ⓘ
probability on dynamical systems ⓘ |
| namedAfter | Mark Kac NERFINISHED ⓘ |
| relatedTo |
Birkhoff ergodic theorem
NERFINISHED
ⓘ
Markov chain stationary distributions ⓘ Poincaré recurrence theorem NERFINISHED ⓘ |
| relates |
expected return time to a set
ⓘ
inverse of the measure of that set ⓘ |
| requires | μ(A) > 0 for the set A ⓘ |
| statement | For a measure-preserving transformation on a probability space, the expected return time to a measurable set A of positive measure equals 1 divided by the measure of A. ⓘ |
| typicalFormulation | If T is a measure-preserving transformation on (X, Σ, μ) and A has μ(A) > 0, then the integral over A of the first return time to A equals 1. ⓘ |
| usedFor |
applications in Markov processes
ⓘ
applications in statistical mechanics ⓘ computing expected recurrence times ⓘ understanding recurrence in ergodic systems ⓘ |
| yearIntroducedApprox | 1940s ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.