Martingale central limit theorem

In probability theory, the central limit theorem says that the sum of many independent identically-distributed random variables, when scaled appropriately, converges in distribution to a standard normal distribution. The martingale central limit theorem generalizes this result to martingales, which are stochastic processes where the change in the value of the process from time t to time t + 1 has expectation zero, even conditioned on previous outcomes.

Here is a simple version of the martingale central limit theorem: Let


 * $$X_1, X_2, \dots\,$$

be a martingale, i.e. suppose


 * $$\operatorname{E}[X_{t+1} - X_t \vert X_1,\dots, X_t]=0.\,$$

Define


 * $$\sigma_t^2 = \operatorname{E}[(X_{t+1}-X_t)^2|X_1, \ldots, X_t],$$

and let


 * $$\tau_v = \min\left\{t : \sum_{i=1}^{t} \sigma_i^2 \ge v\right\}.$$

Then


 * $$\frac{X_{\tau_v}}{\sqrt{v}}$$

converges in distribution to the normal distribution with mean 0 and variance 1.