Break of temporal symmetry in a stationary Markovian setting: evidencing an arrow of time, and parameterizing linear dependencies using fractional low-order joint moments

  • Alin Andrei Carsteanu
  • Andreas LangousisEmail author
Original Paper


We demonstrate that “an arrow of time” that is being determined by the joint distributions of successive process variables, or equivalently a break of temporal symmetry (i.e. a symmetry/asymmetry dichotomy), can be evidenced solely on probabilistic grounds, on the basis of structural dependencies and statistical attributes of observed quantities, without the intervention of any symmetric or asymmetric physical laws. We do so for the simplest case of stable Markovian recursions, and show that a break of temporal symmetry can occur as the combined effect of lack of Gaussianity and statistical dependencies, even in the case when the increments of the generated process are independent and identically distributed with symmetric marginal. This striking result occurs under conditions of stationarity, without any changes in the dynamic recursion equation of the process, allowing for statistical characterization of temporal symmetries versus asymmetries. To that end, we introduce and exemplify the use of an estimator based on fractional low-order joint moments, which exists for all stationary stochastic processes with strictly stable symmetric marginals, and can be used to parameterize their dependence structure in a linear setting.


Arrow of time Linear stochastic process Stable distribution Fractional low-order joint moments Markovian recursion Structural dependence 



The authors would like to dedicate this paper to the memory of two colleagues and friends (in alphabetical order): Alain Arnéodo and Peter Rasmussen who, in different ways, shaped our understanding and modelling of non-linear processes in geophysics. The constructive comments and suggestions of two anonymous reviewers and the Associate Editor are explicitly acknowledged.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.ESFM – Instituto Politécnico NacionalMexico CityMexico
  2. 2.Department of Civil EngineeringUniversity of PatrasPatrasGreece

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