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.

Observed surface forms (2)

Surface form	As subject	As object
Berry–Esseen bounds for dependent variables	0	1
multivariate Berry–Esseen bounds	0	1

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 →

Referenced by (3)

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

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

central limit theorem → relatedTo → Berry–Esseen theorem →