Large deviations for the local fluctuations of random walks and new insights into the "randomness" of Pi

Details Large deviations for the local fluctuations of random walks and new insights into the "randomness" of Pi Large deviations for the local fluctuations of random walks and new insights into the "randomness" of Pi

2010-04-21

arxiv.org/abs/1004.3713v2

Mathematics

Probability

31 pages, 6 figures

Scientific paper

We establish large deviations properties valid for almost every sample path of a class of stationary mixing processes $(X_1,..., X_n,...)$. These properties are inherited from those of $S_n=\sum_{i=1}^nX_i$ and describe how the local fluctuations of almost every realization of $S_n$ deviate from the almost sure behavior. These results apply to the fluctuations of Brownian motion, Birkhoff averages on hyperbolic dynamics, as well as branching random walks. Also, they lead to new insights into the "randomness" of the digits of expansions in integer bases of Pi. We formulate a new conjecture, supported by numerical experiments, implying the normality of Pi.

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