Central limit theorem and why it matters in finance

Give a statement of the central limit theorem and explain what it implies for quantitative finance.

Sums of many independent, finite-variance shocks are approximately normal. If $X_i$ are i.i.d. with mean $\mu$ and variance $\sigma^2<\infty$, then $\sqrt{n}(\bar X_n-\mu)/\sigma$ tends to $N(0,1)$. In finance this justifies normal log returns over longer horizons (hence lognormal prices and Brownian motion as a limit of random walks), and the $1/\sqrt{n}$ error of Monte Carlo pricing. It fails with fat tails, infinite variance or strong dependence.