Martingale central limit theorem

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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 with bounded increments, i.e., suppose

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

and

$|X_{t+1} - X_t| \le k$

almost surely for some fixed bound k and all t. Also assume that $|X_1|\le k$ almost surely.

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 as $v \to +\infty \!$ . More explicitly,

$\lim_{v \to +\infty} \operatorname{P} \left(\frac{X_{\tau_v}}{\sqrt{v}} < x\right) = \Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x \exp\left(-\frac{u^2}{2}\right) \, du, \quad x\in\mathbb{R}.$