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. 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 X_k 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 (i.e. the inverse implication does not hold in general). 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 X_k by using proof by contradiction. This procedure involves proving that Lindeberg's condition fails for X_k.

Interpretation

Because the Lindeberg condition implies as, it guarantees that the contribution of any individual random variable X_k to the variance s_n^2 is arbitrarily small, for sufficiently large values of n.

Example

Consider the following informative example which satisfies the Lindeberg condition. Let \xi_i be a sequence of zero mean, variance 1 iid random variables and a_i a non-random sequence satisfying: Now, define the normalized elements of the linear combination: which satisfies the Lindeberg condition: but \xi_i^2 is finite so by DCT and the condition on the a_i we have that this goes to 0 for every \varepsilon.