  There are non-Gaussian time series that admit a causal linear autoregressive
moving average (ARMA) model when regressing the future on the past, but not
when regressing the past on the future. The reason is that, in the latter case,
the regression residuals are only uncorrelated but not statistically
independent of the future. In previous work, we have experimentally verified
that many empirical time series indeed show such a time inversion asymmetry.
For various physical systems, it is known that time-inversion asymmetries are
linked to the thermodynamic entropy production in non-equilibrium states. Here
we show that such a link also exists for the above unidirectional linearity.
  We study the dynamical evolution of a physical toy system with linear
coupling to an infinite environment and show that the linearity of the dynamics
is inherited to the forward-time conditional probabilities, but not to the
backward-time conditionals. The reason for this asymmetry between past and
future is that the environment permanently provides particles that are in a
product state before they interact with the system, but show statistical
dependencies afterwards. From a coarse-grained perspective, the interaction
thus generates entropy. We quantitatively relate the strength of the
non-linearity of the backward conditionals to the minimal amount of entropy
generation.
