central limit theorem

E4991

probability theorem statistical 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.

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All labels observed (8)

Label	Occurrences
central limit theorem canonical	10
Lyapunov central limit theorem	3
central limit theorem for martingales	2
Central Limit Theorem	1
Lindeberg central limit theorem	1
Lindeberg–Feller central limit theorem	1
classical central limit theorem	1
multivariate central limit theorem	1

Statements (50)

Predicate	Object
instanceOf	probability theorem ⓘ statistical theorem ⓘ
alsoKnownAs	CLT ⓘ
appliesTo	identically distributed random variables ⓘ independent random variables ⓘ
approximationQuality	improves as sample size increases ⓘ
conclusionDistribution	normal distribution ⓘ
describes	convergence in distribution of normalized sums of random variables ⓘ
doesNotRequire	original distribution to be normal ⓘ
failsIf	variance is infinite ⓘ
field	probability theory ⓘ statistics ⓘ
hasApplicationDomain	econometrics ⓘ engineering ⓘ machine learning ⓘ physics ⓘ quantitative finance ⓘ
hasFormulation	convergence of sample mean to normal distribution ⓘ convergence of standardized sum to standard normal distribution ⓘ
hasHistoricalContributor	Abraham de Moivre ⓘ Aleksandr Lyapunov ⓘ Carl Friedrich Gauss ⓘ Jarl Waldemar Lindeberg ⓘ Pierre-Simon Laplace ⓘ
hasVariant	central limit theorem self-linksurface differs ⓘ surface form: Lindeberg central limit theorem central limit theorem self-linksurface differs ⓘ surface form: Lyapunov central limit theorem central limit theorem self-linksurface differs ⓘ surface form: central limit theorem for martingales central limit theorem for triangular arrays ⓘ classical central limit theorem ⓘ functional central limit theorem ⓘ central limit theorem self-linksurface differs ⓘ surface form: multivariate central limit theorem
implies	approximate normality of sample mean for large samples ⓘ
isTypeOf	convergence theorem ⓘ limit theorem ⓘ
relatedTo	Berry–Esseen theorem ⓘ characteristic functions ⓘ law of large numbers ⓘ moment generating functions ⓘ stable distributions ⓘ
requiresCondition	finite mean ⓘ finite variance ⓘ
statesThat	the standardized sum of many independent identically distributed random variables converges in distribution to a normal distribution ⓘ
underpins	asymptotic normality of estimators ⓘ large-sample theory in statistics ⓘ
usedFor	approximating sampling distributions ⓘ construction of confidence intervals ⓘ error analysis in Monte Carlo methods ⓘ hypothesis testing ⓘ normal approximation to Poisson distribution ⓘ normal approximation to binomial distribution ⓘ

Referenced by (20)

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

Brownian motion → relatedConcept → central limit theorem ⓘ

central limit theorem → hasVariant → central limit theorem self-linksurface differs ⓘ

this entity surface form: Lindeberg central limit theorem

central limit theorem → hasVariant → central limit theorem self-linksurface differs ⓘ

this entity surface form: Lyapunov central limit theorem

central limit theorem → hasVariant → central limit theorem self-linksurface differs ⓘ

this entity surface form: multivariate central limit theorem

central limit theorem → hasVariant → central limit theorem self-linksurface differs ⓘ

this entity surface form: central limit theorem for martingales

Gaussian law of error → relatedTo → central limit theorem ⓘ

CLT → fullName → central limit theorem ⓘ

this entity surface form: Central Limit Theorem

CLT → hasVariant → central limit theorem ⓘ

this entity surface form: Lindeberg–Feller central limit theorem

CLT → hasVariant → central limit theorem ⓘ

this entity surface form: Lyapunov central limit theorem

CLT → hasVariant → central limit theorem ⓘ

this entity surface form: central limit theorem for martingales

Berry–Esseen theorem → refines → central limit theorem ⓘ

Berry–Esseen theorem → relatedTo → central limit theorem ⓘ

this entity surface form: Lyapunov central limit theorem

law of large numbers → relatesTo → central limit theorem ⓘ

Probability Theory → usesConcept → central limit theorem ⓘ

subject surface form: Probability theory

Foundations of Probability → topic → central limit theorem ⓘ

Stirling's approximation → relatedTo → central limit theorem ⓘ

Andrew C. Berry → contributedTo → central limit theorem ⓘ

Khinchin's law of the iterated logarithm → refines → central limit theorem ⓘ

Limit Laws for Sums of Independent Random Variables → topic → central limit theorem ⓘ

Limit Laws for Sums of Independent Random Variables → relatedTo → central limit theorem ⓘ

this entity surface form: classical central limit theorem