Central limit theorems and tightness

Let the observations have mean μ and variance σ² strictly between zero and infinity. Their sum has mean nμ and variance nσ². Subtracting the mean removes the deterministic drift, and dividing by σ√n keeps the variance equal to one. The iid central limit theorem states convergence in distribution to a standard normal variable. The equivalent statement for the sample average multiplies its error by √n/σ. Omitting the square root or forgetting to center changes the proposed limit completely.

This theorem concerns fixed continuity thresholds of the limiting normal law. It does not, from finite variance alone, supply a universal finite-sample error rate, accurate extreme-tail probabilities, or a density approximation for every distribution. Quantitative results such as Berry–Esseen require stronger moment assumptions. At lattice thresholds, a continuity correction can improve a normal approximation, but it remains an approximation whose accuracy must be assessed for the actual sample size and distribution.

After standardizing one observation to mean zero and variance one, its characteristic function near zero is one minus t²/2 plus a remainder smaller than t². Independence makes the transform of the normalized sum the nth power of the single-observation transform evaluated at t/√n. The quadratic term accumulates to a finite exponent, while the smaller remainder disappears. The resulting transform exp(−t²/2) identifies the standard normal law through the continuity theorem.

Tightness explains another necessary piece of limit reasoning: normalized mass must not escape arbitrarily far. Here Chebyshev gives a uniform tail bound of one over M² because every Zn has variance one. On the real line, tightness ensures that every subsequence has a further weakly convergent subsequence. Identification then matters: the transform argument forces all such limits to be the same Gaussian law. Tightness supplies subsequences, not their Gaussian identity.

A triangular array allows each row to contain different independent centered variables. Normalize each row so its total variance is one. Lindeberg’s sufficient condition asks that, for every positive ε, the total second-moment contribution of terms larger than ε tends to zero. It prevents a few macroscopic jumps from controlling the row sum. Under this condition the row sums converge to a standard normal law, extending the iid theorem to many unequal small contributions.

Unit variance alone does not establish this condition. A row consisting of one symmetric sign and any number of zero variables has total variance one and is tight, but its sum always has a two-point law. The sign remains a full-size contribution as the row grows. Heavy tails, dependence, or a dominant coordinate can similarly defeat a casual Gaussian claim. Diagnose the assumptions before treating a smooth-looking finite histogram as proof of a central limit theorem.