Isserlis’ theorem in probability theory

E284666

result in mathematical statistics 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.

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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	Gaussian moment theorem ⓘ linked to: Isserlis’ theorem in probability theory Isserlis’ formula ⓘ linked to: Isserlis’ theorem in probability theory
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	Wick’s theorem for Gaussian random variables ⓘ linked to: Isserlis’ theorem in probability theory
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 ⓘ