Bochner theorem on characteristic functions

E613407

mathematical theorem

The Bochner theorem on characteristic functions is a fundamental result in probability theory and harmonic analysis that characterizes which functions are Fourier transforms of probability measures by requiring them to be positive-definite, continuous, and normalized at zero.

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All labels observed (3)

Label	Occurrences
Bochner's theorem	2
Bochner theorem on characteristic functions canonical	1
Lévy–Khintchine formula	1

Statements (42)

Predicate	Object
instanceOf	mathematical theorem ⓘ
appliesTo	functions on locally compact abelian groups ⓘ functions on the real line ⓘ
assumption	function is bounded ⓘ function is continuous at zero ⓘ function is defined on a locally compact abelian group ⓘ function is normalized at zero ⓘ function is positive-definite ⓘ
characterizes	Fourier transforms of finite positive measures ⓘ Fourier transforms of probability measures on the real line ⓘ
codomain	space of finite positive measures ⓘ
concerns	Fourier transforms of probability measures ⓘ characteristic functions ⓘ
conclusion	existence of a probability measure with given characteristic function ⓘ function is the Fourier transform of a unique finite positive measure ⓘ
domain	space of continuous positive-definite functions with value 1 at zero ⓘ
field	harmonic analysis ⓘ probability theory ⓘ
guarantees	existence of a representing measure ⓘ uniqueness of the representing measure ⓘ
hasVersion	Bochner theorem for finite positive measures NERFINISHED ⓘ Bochner theorem for probability measures NERFINISHED ⓘ Bochner theorem on locally compact abelian groups NERFINISHED ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	every characteristic function is bounded by 1 in modulus ⓘ every characteristic function is positive-definite ⓘ every characteristic function is uniformly continuous ⓘ
isToolFor	characterization of probability distributions via characteristic functions ⓘ representation of positive-definite functions as Fourier transforms ⓘ
mathematicalArea	functional analysis ⓘ measure theory ⓘ
namedAfter	Salomon Bochner NERFINISHED ⓘ
normalizationCondition	value at zero equals 1 ⓘ
relatedTo	Fourier–Stieltjes transform NERFINISHED ⓘ Herglotz representation theorem NERFINISHED ⓘ Lévy continuity theorem NERFINISHED ⓘ
usedIn	construction of probability measures from characteristic functions ⓘ harmonic analysis on locally compact abelian groups ⓘ study of infinitely divisible distributions ⓘ
usesConcept	continuity ⓘ normalization at zero ⓘ positive-definite function ⓘ

Referenced by (4)

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

Salomon Bochner → notableFor → Bochner theorem on characteristic functions ⓘ

Wiener–Khinchin theorem → relatedTo → Bochner theorem on characteristic functions ⓘ

this entity surface form: Bochner's theorem

Paul Lévy → knownFor → Bochner theorem on characteristic functions ⓘ

this entity surface form: Lévy–Khintchine formula

Khinchin's representation theorem → isRelatedTo → Bochner theorem on characteristic functions ⓘ

this entity surface form: Bochner's theorem