  Inspired from non-equilibrium statistical physics models, a general framework
enabling the definition and synthesis of stationary time series with a priori
prescribed and controlled joint distributions is constructed. Its central
feature consists of preserving for the joint distribution the simple product
struc- ture it has under independence while enabling to input con- trolled and
prescribed dependencies amongst samples. To that end, it is based on products
of d-dimensional matrices, whose entries consist of valid distributions. The
statistical properties of the thus defined time series are studied in details.
Having been able to recast this framework into that of Hidden Markov Models
enabled us to obtain an efficient synthesis procedure. Pedagogical well-chosen
examples (time series with the same marginal distribution, same covariance
function, but different joint distributions) aim at illustrating the power and
potential of the approach and at showing how targeted statistical prop- erties
can be actually prescribed.
