Large deviations for the local fluctuations of random walks

We establish large deviation properties valid for almost every sample path of a class of stationary mixing processes (X₁,⋯,X _n,⋯). These properties are inherited from those of S _n=∑_i=1ⁿ X_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.