A normal number theorem for Brownian motion
Mathematics Research Reports, Volume 2 (2021), pp. 15-20.

We prove that we can generate “a lot” of random normal numbers (in the sense of full Hausdorff dimension), which are outside the scope of Borel’s Normal Number Theorem (in the sense of zero Lebesgue measure). The ingredient is to run Brownian motion over a specific Cantor-like time-set. These are closely related to an equidistributed property of some dynamical orbit with random inputs, which itself is new and significant.

Accepted:
Published online:
DOI: 10.5802/mrr.7
Classification: 37A05,  11K16,  11K36,  60G15,  60G17
Keywords: Brownian motion, fractional Brownian motion, equidistribution sequence, normal number, dynamical orbit
Narn-Rueih Shieh 1

1 Mathematics Department & Applied Math Institute, Emeritus Room, 4F Astro-Math Building, National Taiwan University, Taipei 10617, Taiwan
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Narn-Rueih Shieh. A normal number theorem for Brownian motion. Mathematics Research Reports, Volume 2 (2021), pp. 15-20. doi : 10.5802/mrr.7. https://mrr.centre-mersenne.org/articles/10.5802/mrr.7/

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