Central Limit Theorem
The central limit theorem says that sums of many independent, finite-variance shocks look Gaussian — which is why so many models start with a normal, and why they fail when those assumptions fail.
Definition
For i.i.d. X_i with finite variance, √n (X̄ − μ) → N(0, σ²). Dependent and heavy-tailed versions exist but require more structure. The CLT is about sums, not about a single day in the market.
Why it matters
It justifies normal approximations for estimators and for some aggregated P&L. It does not justify a normal for tomorrow’s S&P return.
Case
A sum of 250 independent desk errors may look Gaussian. A single jump-diffusion day will not. Using CLT language for daily returns is a category error.
How to read it
Check independence and finite variance. Infinite-variance (stable) shocks do not give you a Gaussian limit.