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

摘要

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