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Inspired from modern out-of-equilibrium statistical physics models, a matrix product based framework is defined and studied, that permits the formal definition of random vectors and 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 probability densities by a product of matrices of probability densities. It is first shown that this matrix product model can be remapped onto the framework of Hidden Markov Models. Second, combining this dual perspective enables us both to study the statistical properties of this model in terms of marginal distributions and dependencies (a stationarity condition is notably devised) and to devise an efficient and accurate numerical synthesis procedure. A design procedure is also described that permits the tuning of model parameters to attain targeted statistical 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 actually be prescribed. [ABSTRACT FROM AUTHOR]