When the Central Limit Theorem Breaks

Frame the CLT as the workhorse behind bell-curve statistics, then surface the two assumptions that quietly do all the work.

• CLT: sums of many small independent draws tend to a Gaussian
• Hidden assumption 1: finite variance of the summands
• Hidden assumption 2: independence (or very weak dependence)

Let the learner pick a distribution type and see the running histogram of partial sums, then commit to whether the result will look Gaussian.

• Choose Gaussian, exponential, or Cauchy tails
• Watch the histogram update with sample size n
• Commit: will this sum look bell-shaped at large n?

Ask the learner to identify which assumption of the CLT is most directly violated by a Cauchy-distributed step, before the evidence scene reveals the answer.

• One focused prediction about variance vs independence

Display three animated histograms side by side: Gaussian steps converge to a bell, heavy-tailed (Pareto) steps collapse into a few giant spikes, and correlated steps drift without concentrating.

• Gaussian i.i.d.: clean bell emerges by n ≈ 100
• Pareto i.i.d. with tail exponent ≤ 2: histograms never stabilize into a bell
• AR(1) correlated steps: width shrinks too slowly and shape leaks the dependence

Walk through the mechanism: heavy tails make variance (and higher moments) infinite, so the √n normalization in the CLT denominator diverges; correlations break independence so the variance of the average no longer shrinks like 1/n.

• Cauchy / alpha-stable: variance is undefined, so no Gaussian limit exists
• Pareto with α ≤ 2: the largest single term dominates the sum, not the average
• Correlated steps: Var(X̄) shrinks slower than 1/n, and the limit shape carries the dependence

Let the learner change the tail exponent and the autocorrelation coefficient, and watch the empirical variance of the mean plotted against sample size on a log-log axis.

• Slope of −1 means healthy 1/n CLT scaling
• Slopes flatter than −1 indicate dependence or infinite variance
• Curves for α < 2 never develop a stable slope

Clarify the exact boundaries: Pareto with α > 2 still works, weakly dependent sequences under mixing conditions still work, and truncated heavy tails are fine.

• Tail exponent α > 2 restores finite variance and recovers the CLT
• Mixing / weak dependence allows a generalized CLT
• Truncating extremes removes the heavy-tail pathology

Transfer the explanation to daily equity returns: a learner adjusts tail exponent and autocorrelation on synthetic return data and watches VaR estimates compared to Gaussian assumptions.

• Equity returns empirically have heavy tails and volatility clustering
• Gaussian VaR systematically underestimates true tail risk
• Identifying the failure mode tells you which model family to use

Resolve the driving question by naming both failure modes in one sentence, then summarize the practical diagnostic: check variance finiteness and check independence.

• Heavy tails → infinite variance → √n normalization collapses
• Correlated steps → dependence → variance of mean no longer scales as 1/n
• Diagnostic: estimate tail exponent and autocorrelation before trusting a Gaussian limit