A combinatorial view of stochastic processes: White noise

Abstract

White noise is a fundamental and fairly well understood stochastic process that conforms to the conceptual basis for many other processes, as well as for the modeling of time series. Here, we push a fresh perspective toward white noise that, grounded on combinatorial considerations, contributes to giving new interesting insights both for modeling and theoretical purposes. To this aim, we incorporate the ordinal pattern analysis approach, which allows us to abstract a time series as a sequence of patterns and their associated permutations, and introduce a simple functional over permutations that partitions them into classes encoding their level of asymmetry. We compute the exact probability mass function (p.m.f.) of this functional over the symmetric group of degree [Formula: see text], thus providing the description for the case of an infinite white noise realization. This p.m.f. can be conveniently approximated by a continuous probability density from an exponential family, the Gaussian, hence providing natural sufficient statistics that render a convenient and simple statistical analysis through ordinal patterns. Such analysis is exemplified on experimental data for the spatial increments from tracks of gold nanoparticles in 3D diffusion.