Unveiling Amplitude Distributions via the Ordinal Language of Random Walks

Abstract

Ordinal patterns are widely used to characterize temporal organization in time series, yet they are often considered insensitive to the amplitude distribution of the data. In this work, we show that this limitation can be overcome by considering the ordinal structure of integrated time series. We investigate random walks generated from independent non-Gaussian increments and derive analytical expressions for the ordinal pattern probabilities associated with them. We show that for symmetric distributions, the probabilities of some ordinal patterns are fully determined by symmetry arguments, while those for the remaining patterns depend explicitly on the shape of the increments distribution. Numerical simulations based on $q$-Gaussian increments validate the theoretical predictions. We further show that the construction of the random walk itself plays a fundamental role in the accurate characterization of non-Gaussian fluctuations, as different centering procedures may significantly affect the resulting ordinal statistics. Finally, we validate the proposed framework using financial time series, showing that the ordinal distributions of integrated logarithmic returns capture non-Gaussian features consistent with a cubic law.