Berry–Esseen theorem

E32545

probability theorem quantitative central limit theorem result in probability theory

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.

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All labels observed (4)

Label	Occurrences
Berry–Esseen theorem canonical	6
Berry–Esseen bounds for dependent variables	1
Esseen’s inequality	1
multivariate Berry–Esseen bounds	1

How this entity was disambiguated

This entity first appeared as the object of triple T243806 — 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: Berry–Esseen theorem
Context triple: [central limit theorem, relatedTo, Berry–Esseen theorem]

A. central limit theorem
The central limit theorem is a fundamental result in probability theory stating that the sum (or average) of many independent, identically distributed random variables tends to follow a normal distribution, regardless of the original variables’ distribution, under mild conditions.
B. Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
C. Gauss–Markov theorem
The Gauss–Markov theorem is a fundamental result in statistics stating that, under certain conditions, the ordinary least squares estimator is the best linear unbiased estimator (BLUE) of the coefficients in a linear regression model.
D. Gaussian law of error
The Gaussian law of error is a fundamental statistical principle stating that measurement errors tend to follow a normal (bell-shaped) distribution, forming the basis of much of probability theory and statistical inference.
E. 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.
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: Berry–Esseen theorem
Target entity description: 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.

A. central limit theorem
The central limit theorem is a fundamental result in probability theory stating that the sum (or average) of many independent, identically distributed random variables tends to follow a normal distribution, regardless of the original variables’ distribution, under mild conditions.
B. Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
C. Gauss–Markov theorem
The Gauss–Markov theorem is a fundamental result in statistics stating that, under certain conditions, the ordinary least squares estimator is the best linear unbiased estimator (BLUE) of the coefficients in a linear regression model.
D. Gaussian law of error
The Gaussian law of error is a fundamental statistical principle stating that measurement errors tend to follow a normal (bell-shaped) distribution, forming the basis of much of probability theory and statistical inference.
E. 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.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	probability theorem ⓘ quantitative central limit theorem ⓘ result in probability theory ⓘ
appliesTo	independent identically distributed random variables ⓘ independent non-identically distributed random variables ⓘ
assumes	finite third absolute moment ⓘ independent random variables ⓘ non-degenerate variance ⓘ
boundDependsOn	number of summands ⓘ third absolute moment ⓘ variance ⓘ
category	limit theorems ⓘ theorems in probability theory ⓘ
concerns	convergence to the normal distribution ⓘ normalized sums of independent random variables ⓘ
errorTermOrder	O(1∕√n) ⓘ
field	mathematical statistics ⓘ probability theory ⓘ
focusesOn	distribution function approximation ⓘ
formalizes	speed of convergence in the central limit theorem ⓘ
gives	explicit constant in CLT approximation ⓘ uniform bound on distribution function distance ⓘ
guarantees	uniform closeness to normal distribution for large n ⓘ
hasApplicationsIn	econometrics ⓘ quantitative finance ⓘ statistics ⓘ stochastic modeling ⓘ
hasGeneralizations	Berry–Esseen theorem self-linksurface differs ⓘ surface form: Berry–Esseen bounds for dependent variables Berry–Esseen theorem self-linksurface differs ⓘ surface form: multivariate Berry–Esseen bounds
involves	third absolute central moment ⓘ variance of summands ⓘ
namedAfter	Andrew C. Berry ⓘ Carl-Gustav Esseen ⓘ
provides	rate of convergence bound ⓘ
refines	central limit theorem ⓘ
relatedTo	Edgeworth expansion ⓘ Lindeberg–Feller central limit theorem ⓘ central limit theorem ⓘ surface form: Lyapunov central limit theorem
statesBoundOfForm	C·ρ3∕σ3√n ⓘ
typeOf	inequality in probability ⓘ limit theorem ⓘ
typicalConstantRange	between 0.4 and 1 in many formulations ⓘ
typicalMetric	Kolmogorov distance ⓘ supremum norm of distribution functions ⓘ
usedFor	accuracy assessment of normal approximation ⓘ error bounds in statistical approximations ⓘ finite-sample analysis in statistics ⓘ probabilistic error estimates in numerical methods ⓘ
yearIntroduced	1941 ⓘ

How these facts were elicited

Referenced by (9)

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

central limit theorem → relatedTo → Berry–Esseen theorem ⓘ

Berry–Esseen theorem → hasGeneralizations → Berry–Esseen theorem self-linksurface differs ⓘ

this entity surface form: multivariate Berry–Esseen bounds

Berry–Esseen theorem → hasGeneralizations → Berry–Esseen theorem self-linksurface differs ⓘ

this entity surface form: Berry–Esseen bounds for dependent variables

Carl-Gustav Esseen → knownFor → Berry–Esseen theorem ⓘ

Carl-Gustav Esseen → notableWork → Berry–Esseen theorem ⓘ

this entity surface form: Esseen’s inequality

Lindeberg–Feller central limit theorem → relatedTo → Berry–Esseen theorem ⓘ

Edgeworth expansion → isRelatedTo → Berry–Esseen theorem ⓘ

Andrew C. Berry → notableFor → Berry–Esseen theorem ⓘ

Andrew C. Berry → coNames → Berry–Esseen theorem ⓘ

All labels observed (4)

How this entity was disambiguated Show

Statements (49)

How these facts were elicited Show

Referenced by (9)

How this entity was disambiguated

How these facts were elicited