the Lindeberg-Feller central limit theorem
/ LIN-de-berg FEL-er /
The classic central limit theorem assumes the summands are identically distributed — every term drawn from the same law. But real sums often mix different pieces: errors of different sizes, contributions from different sources. When can such a sum of independent but NON-identical pieces still be normal in the limit? The Lindeberg-Feller theorem is the definitive answer, replacing the rigid 'identical' assumption with a flexible condition on the summands.
Set it up by triangular array: for each n you have independent terms X_{n,1}, ..., X_{n,n} with mean 0 and variances summing to s_n^2. The Lindeberg condition asks that, after standardizing, no single term contributes a non-vanishing share of the total variance from its large values: for every epsilon, the sum over k of E[ X_{n,k}^2 * (indicator that |X_{n,k}| > epsilon*s_n) ] / s_n^2 approaches 0 as n grows. In plain words: each piece must be uniformly small relative to the whole, so the bell emerges from many comparable contributions rather than one dominant spike. If this holds, the standardized sum converges in distribution to standard normal — no identical-distribution assumption needed.
Feller's contribution makes it sharp and that is the beautiful part: under a mild 'no dominant term' uniform-smallness assumption, the Lindeberg condition is not just sufficient but NECESSARY for the normal limit. So this is the essentially complete characterization of when independent summands aggregate to a bell. The honest takeaway: the normal is the limit precisely when no individual term hogs the variance. If one term dominates — its variance does not shrink relative to the total — the limit can be non-normal, and the CLT story breaks.
Add independent measurements from instruments of varying precision: term k is Normal(0, sigma_k^2) with the sigma_k differing. As long as no single sigma_k^2 stays a fixed fraction of the running total s_n^2 (no instrument dominates), the Lindeberg condition holds and the standardized total is still asymptotically standard normal — different-sized pieces, same bell.
Non-identical independent pieces still give a normal limit — provided no single piece dominates the variance.
The normal limit holds exactly when no single term dominates the total variance (the Lindeberg condition). If one term hogs the variance, the limit need not be normal.