  Discretization of continuous stochastic processes is needed to numerically
simulate them or to infer models from experimental time series. However,
depending on the nature of the process, the same discretization scheme, if not
accurate enough, may perform very differently for the two tasks. Exact
discretizations, which work equally well at any scale, are characterized by the
property of invariance under coarse-graining. Motivated by this observation, we
build an explicit Renormalization Group approach for Gaussian time series
generated by auto-regressive models. We show that the RG fixed points
correspond to discretizations of linear SDEs, and only come in the form of
first order Markov processes or non-Markovian ones. This fact provides an
alternative explanation of why standard delay-vector embedding procedures fail
in reconstructing partially observed noise-driven systems. We also suggest a
possible effective Markovian discretization for the inference of partially
observed underdamped equilibrium processes based on the exploitation of the
Einstein relation.
