Cameron–Martin theorem
E59985
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).
All labels observed (2)
| Label | Occurrences |
|---|---|
| Cameron–Martin theorem canonical | 5 |
| Cameron–Martin space | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T478566 — 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: Cameron–Martin theorem Context triple: [Girsanov theorem, relatedTo, Cameron–Martin theorem]
-
A.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
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B.
Girsanov theorem
Girsanov theorem is a fundamental result in stochastic calculus that describes how the dynamics of stochastic processes, particularly Brownian motion, change under an equivalent change of probability measure.
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C.
Radon–Nikodym derivative
The Radon–Nikodym derivative is a function that represents how one measure changes with respect to another absolutely continuous measure, playing a central role in modern probability theory and measure theory.
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D.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
-
E.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
- 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: Cameron–Martin theorem Target entity description: 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).
-
A.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
B.
Girsanov theorem
Girsanov theorem is a fundamental result in stochastic calculus that describes how the dynamics of stochastic processes, particularly Brownian motion, change under an equivalent change of probability measure.
-
C.
Radon–Nikodym derivative
The Radon–Nikodym derivative is a function that represents how one measure changes with respect to another absolutely continuous measure, playing a central role in modern probability theory and measure theory.
-
D.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
-
E.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in functional analysis ⓘ theorem in probability theory ⓘ |
| appliesTo | Gaussian measures on infinite-dimensional spaces ⓘ |
| characterizes | quasi-invariance of Gaussian measures under translations ⓘ |
| concernsProperty |
absolute continuity of shifted Gaussian measures
ⓘ
mutual singularity of measures ⓘ quasi-invariance under translations ⓘ |
| describes | change of Gaussian measures under shifts ⓘ |
| field |
functional analysis
ⓘ
measure theory ⓘ probability theory ⓘ |
| gives | explicit formula for Radon–Nikodym derivative of shifted Gaussian measure ⓘ |
| hasComponent | definition of Cameron–Martin space as reproducing kernel Hilbert space of Gaussian measure ⓘ |
| hasConsequence |
characterization of admissible shifts of Gaussian paths
ⓘ
description of support of Gaussian measures on Banach spaces ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| holdsFor |
centered Gaussian measures
ⓘ
non-degenerate Gaussian measures ⓘ |
| involves |
Cameron–Martin theorem
self-linksurface differs
ⓘ
surface form:
Cameron–Martin space
Gaussian measure ⓘ Hilbert spaces ⓘ Radon–Nikodym derivative ⓘ separable Banach spaces ⓘ translation operator ⓘ |
| isAbout |
Cameron–Martin Hilbert subspace
ⓘ
shift invariance properties of Gaussian measures ⓘ structure of Gaussian measures on Banach spaces ⓘ |
| mathematicalDomain |
Gaussian measure theory
ⓘ
infinite-dimensional analysis ⓘ |
| namedAfter |
Richard H. Cameron
ⓘ
W. T. Martin ⓘ |
| relatedTo |
Gaussian process
ⓘ
Girsanov theorem ⓘ Hilbert space embedding of Cameron–Martin space ⓘ Wiener measure ⓘ abstract Wiener space ⓘ |
| statesThat |
translation by a vector outside the Cameron–Martin space makes the Gaussian measure mutually singular with the original
ⓘ
translation by an element of the Cameron–Martin space preserves equivalence class of a Gaussian measure ⓘ |
| usedIn |
Brownian motion analysis
ⓘ
Gaussian processes theory ⓘ Malliavin calculus ⓘ infinite-dimensional integration ⓘ large deviations theory ⓘ path space measures ⓘ stochastic analysis ⓘ theory of abstract Wiener spaces ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cameron–Martin theorem Description of subject: 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).
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Cameron–Martin space
this entity surface form:
Cameron–Martin space