Lindeberg’s central limit theorem

Theorem (Lindeberg’s central limit theoremMathworldPlanetmath)

Let X1,X2, be independentPlanetmathPlanetmath random variablesMathworldPlanetmath with distribution functionsMathworldPlanetmath F1,F2,, respectively, such that EXn=μn and VarXn=σn2<, with at least one σn>0. Let


Then the normalized partial sums Sn-ESnsn converge in distribution ( to a random variable with normal distributionMathworldPlanetmath N(0,1) (i.e. the normal convergence holds,) if the following Lindeberg condition is satisfied:


Corollary 1 (Lyapunov’s central limit theorem)

If the Lyapunov condition


is satisfied for some δ>0, the normal convergence holds.

Corollary 2

If X1,X2, are identically distributed random variables, EXn=μ and VarSn=σ2, with 0<σ<, then the normal convergence holds; i.e. Sn-nμσn converges in distribution ( to a random variable with distribution N(0,1).

Reciprocal (Feller)

The reciprocal of Lindeberg’s central limit theorem holds under the following additional assumption:


Historical remark

Title Lindeberg’s central limit theorem
Canonical name LindebergsCentralLimitTheorem
Date of creation 2013-03-22 13:14:25
Last modified on 2013-03-22 13:14:25
Owner Koro (127)
Last modified by Koro (127)
Numerical id 19
Author Koro (127)
Entry type Theorem
Classification msc 60F05
Synonym Lyapunov’s central limit theorem
Synonym central limit theorem
Synonym lyapunov condition
Synonym lindeberg condition
Related topic TightAndRelativelyCompactMeasures
Defines normal convergence
Defines liapunov’s central limit theorem
Defines liapunov condition