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Martingale central limit theorem
From Wikipedia, the free encyclopedia
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
be a martingale, i.e. suppose
![\operatorname{E}[X_{t+1} - X_t \vert X_1,\dots, X_t]=0.\,](../../../math/b/8/8/b885fba7d9be861b3c58ea386e391d3b.png)
Define
![\sigma_t^2 = \operatorname{E}[(X_{t+1}-X_t)^2|X_1, \ldots, X_t],](../../../math/c/e/4/ce478d1f96aa30870a064ee2811eb961.png)
and let
Then
converges in distribution to the normal distribution with mean 0 and variance 1.
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.