the Lindeberg-Feller theorem
/ LIN-de-berg FEL-er /
The classical CLT assumes the summands are independent AND identically distributed. Real applications often violate the identical part: terms have different distributions, variances, and scales. The Lindeberg-Feller theorem is the definitive CLT for independent but not-necessarily-identical summands, and it pins down the EXACT condition under which a Gaussian limit holds — neither too strong nor too weak.
Set it up with a triangular array: for each n, X_{n,1}, ..., X_{n,r_n} are independent, mean 0, with variances sigma_{n,k}^2 summing to s_n^2 = sum_k sigma_{n,k}^2. The Lindeberg condition is: for every epsilon > 0, (1/s_n^2) sum_k E[ X_{n,k}^2 ; |X_{n,k}| > epsilon s_n ] -> 0 as n -> infinity. The theorem says: if the array satisfies the Lindeberg condition then (1/s_n) sum_k X_{n,k} -> N(0,1) in distribution. Feller's converse makes it sharp: provided the array is uniformly asymptotically negligible (no single term dominates, i.e. max_k sigma_{n,k}^2/s_n^2 -> 0), the Lindeberg condition is NECESSARY as well as sufficient for both the CLT and that negligibility. The Lindeberg condition is a uniform smallness requirement: it says the contribution to the total variance coming from large values (beyond epsilon s_n) is asymptotically negligible — no single summand carries non-vanishing weight.
Lindeberg-Feller matters because it is the working CLT for regression residuals, weighted sums, m-dependent sequences (after blocking), and martingale-difference settings (its generalization). It is also conceptually decisive: combined with Feller's converse it shows the Gaussian limit holds essentially exactly when no single term dominates and large deviations of individual terms are uniformly negligible. Honest caveats: the negligibility hypothesis cannot be dropped — if one summand carries a fixed fraction of the variance, the limit is a convolution with that term's law, not Gaussian. And Lindeberg gives only the Gaussian limit; when the array fails negligibility or has heavy tails you are back in the general central limit problem with possibly stable or infinitely-divisible limits.
Take X_{n,k} independent, uniform on [-k, k] for k = 1, ..., n. The variances grow but no single term dominates the total once normalized, and one checks the Lindeberg condition holds, so the normalized sum is asymptotically normal. By contrast, if one term had a fixed fraction of the variance (e.g. one Bernoulli with constant scale), negligibility fails and the limit is not Gaussian.
Lindeberg condition (uniform smallness of tail contributions) + negligibility <=> Gaussian limit, by Feller's converse.
Without uniform asymptotic negligibility (no single term dominates), the limit need not be Gaussian — a dominant term makes the limit a convolution with that term's law.