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

TL;DR

This paper develops large deviation principles for local fluctuations in various stochastic processes and explores their implications for the randomness and normality of Pi's digits.

Contribution

It introduces large deviations results for stationary mixing processes and applies them to diverse systems, providing new insights into Pi's digit randomness.

Findings

Large deviations principles hold for almost all sample paths.

Results apply to Brownian motion, hyperbolic dynamics, and branching random walks.

A new conjecture suggests Pi's digits are normal, supported by numerical evidence.

Abstract

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.