Lindeberg's condition


In probability theory, Lindeberg's condition is a sufficient condition for the central limit theorem to hold for a sequence of independent random variables. 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.

Statement

Let be a probability space, and, be independent random variables defined on that space. Assume the expected values and variances exist and are finite. Also let
If this sequence of independent random variables satisfies Lindeberg's condition:
for all, where 1 is the indicator function, then the central limit theorem holds, i.e. the random variables
converge in distribution to a standard normal random variable as
Lindeberg's condition is sufficient, but not in general necessary.
However, if the sequence of independent random variables in question satisfies
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. Letting and for simplicity, the theorem states
This theorem can be used to disprove the central limit theorem holds for by using proof by contradiction. This procedure involves proving that Lindeberg's condition fails for.

Interpretation

Because the Lindeberg condition implies as, it guarantees that the contribution of any individual random variable to the variance is arbitrarily small, for sufficiently large values of.