Martingale Central Limit Theorem - Statement

Statement

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

-- be a martingale with bounded increments, i.e., suppose

and

almost surely for some fixed bound k and all t. Also assume that almost surely.

Define

and let

Then

converges in distribution to the normal distribution with mean 0 and variance 1 as . More explicitly,

\lim_{\nu \to +\infty} \operatorname{P} \left(\frac{X_{\tau_\nu}}{\sqrt{\nu}} < 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}.

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