Central Limit Theorem Breakdown: Heavy-Tailed Convergence

Source Distribution

⚠ This source has infinite variance. The classical CLT does not apply — the √n-standardized histogram will keep spreading instead of settling into a bell curve.

Sampling

Sample size n 30

Draw speed 400/s

√n CLT standardization

Live Statistics

Sample means drawn

Off-axis rate

0.0%

Est. spread (std dev)

—

Excess kurtosis

—

How it works

The classical (Lindeberg–Lévy) Central Limit Theorem says: for i.i.d. draws X₁…Xₙ with finite mean μ and finite variance σ², the standardized sample mean converges in distribution to a standard normal, no matter how skewed the source is:

Z_n = √n · (X̄_n − μ) / σ   →   N(0, 1)   as n → ∞

Every source here is standardized this same way each time a new sample mean is drawn, and the result is binned into a live density histogram (bars) next to the target N(0,1) curve (line) so you can watch the fit tighten as n grows.

The finite-variance sources — Gaussian, Exponential, Pareto α=3 — all obey the theorem: the bars hug the bell curve more closely as n increases, even though Exponential and Pareto α=3 are themselves heavily skewed.

The infinite-variance sources — Pareto α=1.5 and Cauchy — are where the theorem's finite-variance assumption is not met. A Pareto tail with index α<2 has E[X²]=∞, and Cauchy has no finite mean at all. The generalized CLT for such sources gives a different, correct scaling:

Sum of n draws ~ n^(1/α), not n^(1/2)
so √n·(X̄_n − center) grows like n^(1/α − 1/2) → ∞  for α < 2

Source buttons — pick the distribution the samples are drawn from (Pareto α=3 has finite variance; α=1.5 does not; Cauchy has neither finite mean nor variance).
Sample size n — how many raw draws are averaged into one sample mean before it's plotted.
Draw speed — how many fresh sample means are generated per second.
√n standardization toggle — off shows the raw (μ,σ-normalized) sample mean, which shrinks toward 0 as n grows for any of these sources (a weaker law-of-large-numbers effect); on applies the exact CLT scaling above.
Off-axis rate — the fraction of standardized values landing outside the fixed [-6,6] display window. For finite-variance sources this rate falls toward 0 as n grows; for infinite-variance sources it keeps climbing instead — direct visual evidence that the histogram is not settling down.

Real-world relevance: this finite-variance requirement is why sampling means, confidence intervals and control charts work reliably for measurement noise or transaction amounts, but break down for internet traffic bursts, insurance claim sizes, and other heavy-tailed processes — practitioners in those fields use stable-distribution or extreme-value models instead of a plain Gaussian confidence interval.

Sample-mean density (histogram) Target N(0,1) curve

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