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TL;DR

States that the sum (or average) of a large number of independent, identically distributed random variables with finite variance converges in distribution to a normal distribution, regardless of the original distribution.

By Valenke Exam Prep Team·Last updated 2026-06-03

Central Limit Theorem

States that the sum (or average) of a large number of independent, identically distributed random variables with finite variance converges in distribution to a normal distribution, regardless of the original distribution.

Why it matters for interviews

Justifies the widespread use of normal distributions in finance, underpins confidence interval construction, and explains why many portfolio return distributions appear approximately Gaussian over short horizons.

Definition and Mathematical Foundation

States that the sum (or average) of a large number of independent, identically distributed random variables with finite variance converges in distribution to a normal distribution, regardless of the original distribution.

More precisely, \( \frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} N(0,1) \). The rate of convergence is governed by the Berry-Esseen theorem: the CDF error is bounded by \( C \cdot E[|X-\mu|^3]/(\sigma^3 \sqrt{n}) \) where \( C \leq 0.4748 \). For symmetric distributions, convergence is faster.

Application in Quantitative Finance

Justifies the widespread use of normal distributions in finance, underpins confidence interval construction, and explains why many portfolio return distributions appear approximately Gaussian over short horizons.

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Frequently Asked Questions

What are the conditions for the CLT to hold?▼
The classic Lindeberg-Levy CLT requires iid random variables with finite mean and variance. Extensions (Lindeberg, Lyapunov) relax the identical distribution requirement but still need independence and variance conditions.
Why does the CLT matter for risk management?▼
VaR and other risk metrics often assume normality. The CLT justifies this for aggregated returns over many positions, but the convergence rate matters -- fat-tailed assets may need very large samples before normality kicks in.
How fast does CLT convergence occur?▼
The Berry-Esseen theorem bounds the convergence rate at \( O(1/\sqrt{n}) \). For highly skewed or heavy-tailed distributions, convergence can be slow, which is why quant finance supplements CLT with extreme value theory.