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.

Aliases (6)

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 → surface form: "Lindeberg central limit theorem" central limit theorem → surface form: "Lyapunov central limit theorem" central limit theorem → surface form: "central limit theorem for martingales" central limit theorem for triangular arrays → classical central limit theorem → functional central limit theorem → central limit theorem → 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 (12)

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

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"

central limit theorem → hasVariant → central limit theorem →

this entity surface form: "Lindeberg central limit theorem"

central limit theorem → hasVariant → central limit theorem →

this entity surface form: "Lyapunov central limit theorem"

central limit theorem → hasVariant → central limit theorem →

this entity surface form: "multivariate central limit theorem"

central limit theorem → hasVariant → central limit theorem →

this entity surface form: "central limit theorem for martingales"

Berry–Esseen theorem → refines → central limit theorem →

Brownian motion → relatedConcept → central limit theorem →

Berry–Esseen theorem → relatedTo → central limit theorem →

this entity surface form: "Lyapunov central limit theorem"

Gaussian law of error → relatedTo → central limit theorem →