1.4
Central Limit Theorem
Why does adding up many small, independent shocks always produce a bell curve — whatever the shocks look like?
Goal: watch the standardised sum Zₙ = Sₙ/√n approach N(0,1) for different step laws, and measure how fast the gap closes.
n = 1
Empirical densityN(0,1) density
The coin's sum lives on a lattice, so small n shows spikes; the continuous laws smooth out sooner. Only mean and variance survive the limit.
Berry–Esseen boundMeasured KS distance
Log–log axes: a 1/√n rate is a straight line of slope −½, like the dashed Berry–Esseen bound. The dotted line (0.012) is the sampling noise floor — below it the finite sample, not the sum, sets the distance.