When does the central limit theorem fail?

Question

The central limit theorem promises that a sample mean is approximately Gaussian for large $n$ . But it rests on assumptions that real financial data often violate.

Under what conditions does the CLT fail or mislead, and what happens to the sample mean when it does?

Accepted Answer

The standard CLT needs three things: independent (or weakly dependent) samples, identical distribution, and a finite variance. Break any of them and the clean Gaussian conclusion can collapse. 1. Infinite or ill-defined variance (heavy tails). If the underlying distribution has no finite variance, the CLT simply does not apply. The classic example is the Cauchy distribution: the average of n Cauchy draws is itself Cauchy, no matter how large n is, so the sample mean never concentrates and never becomes Gaussian. Financial returns often have tails heavy enough that variance is effectively unst…

When does the central limit theorem fail?

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