Isserlis’ theorem in probability theory
E284666
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Gaussian moment theorem | 1 |
| Isserlis’ formula | 1 |
| Isserlis’ theorem in probability theory canonical | 1 |
| Wick’s theorem for Gaussian random variables | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2631076 — 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: Isserlis’ theorem in probability theory Context triple: [Wick’s theorem, relatedTo, Isserlis’ theorem in probability theory]
-
A.
Modern Probability Theory and Its Applications
"Modern Probability Theory and Its Applications" is a foundational textbook by Emanuel Parzen that systematically develops modern probability theory and demonstrates its use in a wide range of statistical and applied contexts.
-
B.
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.
-
C.
Khinchin–Kahane type inequalities
Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
-
D.
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.
-
E.
Lindeberg–Feller central limit theorem
The Lindeberg–Feller central limit theorem is a general form of the central limit theorem that provides conditions under which sums of independent, not necessarily identically distributed random variables converge in distribution to a normal law.
- 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: Isserlis’ theorem in probability theory Target entity description: 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.
-
A.
Modern Probability Theory and Its Applications
"Modern Probability Theory and Its Applications" is a foundational textbook by Emanuel Parzen that systematically develops modern probability theory and demonstrates its use in a wide range of statistical and applied contexts.
-
B.
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.
-
C.
Khinchin–Kahane type inequalities
Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
-
D.
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.
-
E.
Lindeberg–Feller central limit theorem
The Lindeberg–Feller central limit theorem is a general form of the central limit theorem that provides conditions under which sums of independent, not necessarily identically distributed random variables converge in distribution to a normal law.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
result in mathematical statistics
ⓘ
theorem in probability theory ⓘ |
| alsoKnownAs |
Isserlis’ theorem in probability theory
ⓘ
surface form:
Gaussian moment theorem
Isserlis’ theorem in probability theory ⓘ
surface form:
Isserlis’ formula
|
| appliesTo |
jointly Gaussian random variables
ⓘ
multivariate normal distributions ⓘ |
| assumption |
finite second moments
ⓘ
joint normality of the variables ⓘ |
| category |
theorems about Gaussian distributions
ⓘ
theorems about moments ⓘ |
| equivalentTo |
Isserlis’ theorem in probability theory
self-linksurface differs
ⓘ
surface form:
Wick’s theorem for Gaussian random variables
|
| field |
Gaussian theory
ⓘ
mathematical statistics ⓘ probability theory ⓘ |
| generalizationOf |
expression of fourth-order moments via covariances
ⓘ
formula for the fourth moment of a Gaussian variable ⓘ |
| implies | all information about Gaussian distributions is contained in first and second moments ⓘ |
| involvesOperation |
pairwise partitioning of indices
ⓘ
summing products of covariances over all pairings ⓘ |
| namedAfter | Leon Isserlis ⓘ |
| originalPublicationLanguage | English ⓘ |
| originalPublicationVenue | Biometrika ⓘ |
| property |
expresses any even-order joint moment as a sum over pairings of indices
ⓘ
moment expression depends only on means and covariances ⓘ odd-order joint moments of centered Gaussian variables are zero ⓘ provides closed-form expressions for Gaussian moments ⓘ |
| publicationYear | 1918 ⓘ |
| relatesConcept |
central moments
ⓘ
covariance ⓘ cumulants ⓘ even-order moments ⓘ higher-order moments ⓘ moment generating functions ⓘ odd-order moments ⓘ pairwise covariances ⓘ raw moments ⓘ |
| usedFor |
computing expectations of products of Gaussian variables
ⓘ
computing moments in multivariate normal distributions ⓘ deriving covariance structures ⓘ simplifying calculations in Gaussian models ⓘ |
| usedIn |
Gaussian process modeling
ⓘ
financial mathematics ⓘ machine learning ⓘ quantum field theory ⓘ statistical signal processing ⓘ time series analysis ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Isserlis’ theorem in probability theory Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
Isserlis’ theorem in probability theory
→
equivalentTo
→
Isserlis’ theorem in probability theory
self-linksurface differs
ⓘ
subject surface form:
Isserlis’ theorem
this entity surface form:
Wick’s theorem for Gaussian random variables
subject surface form:
Isserlis’ theorem
this entity surface form:
Isserlis’ formula
subject surface form:
Isserlis’ theorem
this entity surface form:
Gaussian moment theorem