  Inspired from modern out-of-equilibrium statistical physics models, a matrix
product based framework permits the formal definition of random vectors (and
random time series) whose desired joint distributions are a priori prescribed.
Its key feature consists of preserving the writing of the joint distribution as
the simple product structure it has under independence, while inputing
controlled dependencies amongst components: This is obtained by replacing the
product of distributions by a product of matrices of distributions. The
statistical properties stemming from this construction are studied
theoretically: The landscape of the attainable dependence structure is
thoroughly depicted and a stationarity condition for time series is notably
obtained. The remapping of this framework onto that of Hidden Markov Models
enables us to devise an efficient and accurate practical synthesis procedure. A
design procedure is also described permitting the tuning of model parameters to
attain targeted properties. Pedagogical well-chosen examples of times series
and multivariate vectors aim at illustrating the power and versatility of the
proposed approach and at showing how targeted statistical properties can be
actually prescribed.
