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 (8)

Label	Occurrences
Bochner's theorem	2
Bochner theorem for finite positive measures	1
Bochner theorem for probability measures	1
Bochner theorem on characteristic functions canonical	1
Bochner theorem on locally compact abelian groups	1
Bochner’s theorem	1
Bochner’s theorem in harmonic analysis	1
Lévy–Khintchine formula	1

How this entity was disambiguated

This entity first appeared as the object of triple T6716265 — 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: Bochner theorem on characteristic functions
Context triple: [Salomon Bochner, notableFor, Bochner theorem on characteristic functions]

A. Isserlis’ theorem in probability theory
Isserlis’ theorem in probability theory is a result that expresses higher-order moments of jointly Gaussian random variables in terms of sums of products of their pairwise covariances.
B. Khinchin's representation theorem
Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
C. Wiener–Khinchin theorem
The Wiener–Khinchin theorem is a fundamental result in signal processing and probability theory that relates a wide-sense stationary random process’s autocorrelation function to its power spectral density via the Fourier transform.
D. Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
E. Tauberian theorems
Tauberian theorems are results in mathematical analysis that connect the behavior of transformed series or integrals (such as those summed by Abel or Cesàro methods) back to the asymptotic behavior or convergence of the original sequences or series.
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: Bochner theorem on characteristic functions
Target entity description: 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.

A. Isserlis’ theorem in probability theory
Isserlis’ theorem in probability theory is a result that expresses higher-order moments of jointly Gaussian random variables in terms of sums of products of their pairwise covariances.
B. Khinchin's representation theorem
Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
C. Wiener–Khinchin theorem
The Wiener–Khinchin theorem is a fundamental result in signal processing and probability theory that relates a wide-sense stationary random process’s autocorrelation function to its power spectral density via the Fourier transform.
D. Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
E. Tauberian theorems
Tauberian theorems are results in mathematical analysis that connect the behavior of transformed series or integrals (such as those summed by Abel or Cesàro methods) back to the asymptotic behavior or convergence of the original sequences or series.
F. None of above. chosen

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 ⓘ linked to: Bochner theorem on characteristic functions Bochner theorem for probability measures ⓘ linked to: Bochner theorem on characteristic functions Bochner theorem on locally compact abelian groups ⓘ linked to: Bochner theorem on characteristic functions
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 ⓘ
normalizationCondition	value at zero equals 1 ⓘ
relatedTo	Fourier–Stieltjes transform ⓘ Herglotz representation theorem ⓘ linked to: Herglotz's theorem Lévy continuity theorem ⓘ linked to: Lévy’s continuity theorem
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 ⓘ

How these facts were elicited