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I am teaching a stochastic processes course to MA stat students, and to stay on topic I would like some examples of limiting distributions in stat that are identified as functions of brownian motion or stochastic integrals. I have thought of kolmogorov-smirnov & dickey-fuller, neither of which uses stochastic integration. Examples have to be short, especially, not require lots of stats backround.

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I think you might be interested in the martingale central limit theorem. –  Stéphane Laurent Jul 18 '12 at 14:37
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