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

Abstract

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.