Generalized Central Limit Theorem

Statement¶

The ordinary central limit theorem uses finite variance and normalizes sums by $\sqrt{n}$. The substantive generalized central limit theorem is a classification theorem: a nondegenerate distribution can occur as a weak limit of centered and normalized iid sums if and only if it is stable. In other words, if there are constants $a_n>0$ and $b_n$ such that

for a nondegenerate limit $Z$, then $Z$ must be stable; conversely, stable laws arise as such limits for suitable iid summands. The Gaussian law is the stable case $\alpha=2$. The heavy-tailed stable cases have index $0<\alpha<2$.

For a two-sided regularly varying sufficient condition, assume

where $L$ is slowly varying at infinity, with tail balance

Choose $a_n$ so that $n\mathbb P(|X|>a_n)\to1$. Then, with suitable centering constants $b_n$,

where $Z_\alpha$ is an $\alpha$-stable law whose skewness is determined by $p-q$. For nonnegative Pareto-type examples this reduces to a strongly right-skewed stable limit, and the scaling is of order $n^{1/\alpha}$ up to a slowly varying factor. When $1<\alpha<2$, the mean exists but the variance is infinite, so a common centering choice is $b_n=n\mathbb E[X]$.

A common centering summary is

Here $S_n=\sum_{i=1}^n X_i$ is the sum, $M_n=\max_{1\le i\le n}X_i$ is the maximum, $\mathbb E$ is expectation, and $\Rightarrow$ denotes convergence in distribution. The tail exponent $\alpha$, norming constants $a_n$, centering constants $b_n$, tail-balance weights $p,q$, and stable limit $Z_\alpha$ are defined above. Shared notation is planned.

We record the stable-limit contrast needed by the site’s heavy-tail pages. We do not classify every possible domain of attraction or every parameterization of stable laws.

The classification and sufficient tail condition above are cited from Feller 1971, 2nd ed., Wiley and Resnick 2007; the local argument below addresses only exact Pareto maximum scaling.

Why $\sqrt{n}$ fails¶

The normal law is not the only possible attractor for sums. It is the finite-variance attractor. For the regularly varying tails with $0<\alpha<2$ considered here, the largest observations remain visible at the scale of the centered sum, and $\sqrt{n}$ is no longer the right normalization.

For a Pareto-type tail with exponent $\alpha<2$, the natural scale of the maximum is about $n^{1/\alpha}$. Stable normalization puts the centered sum on that same order. This is why infinite-variance sums can keep producing large jumps instead of smoothing into Gaussian-looking noise.

Scaling argument¶

The full generalized central limit theorem is a stable-law classification result, so we cite it rather than reproduce the proof. What we prove directly is the exact Pareto maximum scaling calculation, which shows why the usual $\sqrt n$ scale is too small when $\alpha<2$.

For a Pareto Type I variable with survival $\bar F(x)=(x_m/x)^\alpha$, choose $a_n=x_m n^{1/\alpha}$. Then for $y>0$ and sufficiently large $n$ such that $a_n y\ge x_m$,

The maximum remains of order $a_n$. Since $a_n$ grows faster than $\sqrt{n}$ when $\alpha<2$, the Gaussian finite-variance scaling is not the right asymptotic scale for such tails. This maximum calculation identifies the right order of extreme observations; it is a scaling heuristic, not a proof of stable convergence for sums.

Pareto scaling comparison¶

Take exact Pareto Type I with $x_m=1$ and $\alpha=1.5$. Its mean is 3 and its variance is infinite. With $S_n=\sum_{i=1}^n X_i$, compare

The identity holds for every sample. Consequently, whenever its interquartile range is nonzero, the spread of $G_n$ is exactly $n^{1/6}$ times that of $T_n$. Here are the normalization factors, rounded to three decimals:

The cited stable-limit theorem implies convergence of the central quantiles of $T_n$ to those of a nondegenerate continuous stable law. Thus the central spread of $G_n$ grows on the $n^{1/6}$ scale. The table evaluates the normalizations; it is neither simulated evidence nor a proof of stable convergence.

Caveats¶

• Stable-law parameterizations vary across books and software. We use only the stable index $\alpha$ and avoid detailed skew/location notation.

• The displayed regular-variation and tail-balance condition covers the non-Gaussian stable range $0<\alpha<2$; the Gaussian boundary needs a different criterion.

• Right-tail regular variation alone is not enough for a general two-sided variable; the left tail can change the scale or skewness of the limit.

• Some infinite-variance distributions can still be in the Gaussian domain of attraction when their truncated second moment is slowly varying.

• For $0<\alpha<1$, the absolute first moment is infinite. At $\alpha=1$, its finiteness depends on the slowly varying factor: the tail-integral criterion is $\int^\infty L(x)/x\,dx<\infty$. For example, a nonnegative variable with survival $1/[x(\log x)^2]$ for $x\ge e$ has a finite mean, since that tail integral equals 1. Exact Pareto tails have infinite mean at $\alpha=1$. See the boundary discussion in Karamata’s theorem. The truncated centering above remains a standard choice at this boundary; finite mean alone does not justify replacing it by $n\mathbb E[X]$ without checking the resulting location shift on the $a_n$ scale.

• Finite samples can look calmer than the asymptotic theory suggests until a large observation arrives.

References¶

• Feller, An Introduction to Probability Theory and Its Applications, Vol. II 1971, 2nd ed., Wiley.

• Resnick, Heavy-Tail Phenomena 2007.

• Taleb, Statistical Consequences of Fat Tails 2020.

Backlinks¶

• Depends on: Regular Variation and Pareto Moment Existence.

• Related to: LLN Failure Under Infinite Mean.

• Used by: Pre-Asymptotic LLN Behavior (planned).

Source and adaptation¶

Adapted from incerto-wiki, content/concepts/theorems/generalized-central-limit-theorem.md, revision 9717c9c (2026-09-13 import). Copyright (c) 2023 xshi19. Licensed under MIT. Links and notation were adapted for this site; executable figures and simulations were replaced with static calculations. No upstream execution or formal-proof verification is claimed for this adaptation.