Lindeberg's condition

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In probability theory, Lindeberg's condition is a sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independent random variables.[1][2][3] Unlike the classical CLT, which requires that the random variables in question have finite variance and be both independent and identically distributed, Lindeberg's CLT only requires that they have finite variance, satisfy Lindeberg's condition, and be independent. It is named after the Finnish mathematician Jarl Waldemar Lindeberg.[4]

Statement

Let (Ω,,) be a probability space, and Xk:Ω,k, be independent random variables defined on that space. Assume the expected values 𝔼[Xk]=μk and variances Var[Xk]=σk2 exist and are finite. Also let sn2:=k=1nσk2.

If this sequence of independent random variables Xk satisfies Lindeberg's condition:

limn1sn2k=1n𝔼[(Xkμk)2𝟏{|Xkμk|>εsn}]=0

for all ε>0, where 1{…} is the indicator function, then the central limit theorem holds, i.e. the random variables

Zn:=k=1n(Xkμk)sn

converge in distribution to a standard normal random variable as n.

Lindeberg's condition is sufficient, but not in general necessary (i.e. the inverse implication does not hold in general). However, if the sequence of independent random variables in question satisfies

maxk=1,,nσk2sn20, as n,

then Lindeberg's condition is both sufficient and necessary, i.e. it holds if and only if the result of central limit theorem holds.

Remarks

Feller's theorem

Feller's theorem can be used as an alternative method to prove that Lindeberg's condition holds.[5] Letting Sn:=k=1nXk and for simplicity 𝔼[Xk]=0, the theorem states

if ε>0, limnmax1knP(|Xk|>εsn)=0 and Snsn converges weakly to a standard normal distribution as n then Xk satisfies the Lindeberg's condition.


This theorem can be used to disprove the central limit theorem holds for Xk by using proof by contradiction. This procedure involves proving that Lindeberg's condition fails for Xk.

Interpretation

Because the Lindeberg condition implies maxk=1,,nσk2sn20 as n, it guarantees that the contribution of any individual random variable Xk (1kn) to the variance sn2 is arbitrarily small, for sufficiently large values of n.

Example

Consider the following informative example which satisfies the Lindeberg condition. Let ξi be a sequence of zero mean, variance 1 iid random variables and ai a non-random sequence satisfying:

maxin|ai|ai20

Now, define the normalized elements of the linear combination:

Xn,i=aiξia2

which satisfies the Lindeberg condition:

in𝔼[|Xi|21(|Xi|>ε)]in𝔼[|Xi|21(|ξi|>εa2maxin|ai|)]=𝔼[|ξi|21(|ξi|>εa2maxin|ai|)]

but ξi2 is finite so by DCT and the condition on the ai we have that this goes to 0 for every ε.

See also

References