Summary
Using self-similarity of Brownian motion and its representation as a product measure on a binary tree, we construct a random sequence of probability measures which converges to the distribution of the Brownian bridge. We establish a large deviation principle for random fields on a binary tree. This leads to a class of probability measures with a certain self-similarity property. The same construction can be carried out forC[0, 1]-valued processes and we can describe, for instance, aC[0, 1]-valued Ornstein-Uhlenbeck process as a large deviation of Brownian sheet.
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Gantert, N. Self-similarity of Brownian motion and a large deviation principle for random fields on a binary tree. Probab. Th. Rel. Fields 98, 7–20 (1994). https://doi.org/10.1007/BF01311346
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DOI: https://doi.org/10.1007/BF01311346
Mathematics Subject Classifications (1991)
- 60F10
- 60J65