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 expected 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_t^2 \ge v\right\}.

Then

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

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

[edit] References

Many other variants on the martingale CLT can be found in:

  • Hall, Peter; and C. C. Heyde (1980). Martingale Limit Theory and Its Application. New York: Academic Press. ISBN 0-12-319350-8. 
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