Here we study the problem of constructing multivariate finite Markov chains whose coordinates are finite univariate Markov chains with given generator matrices. Specifically, we will be concerned here with construction of strong and weak Markov chain structures for a collection of finite Markov chains. We will use methods that are specific for Markov chains, and that are based on the results derived in Chapter 3. In this chapter we shall additionally be concerned with constructing weak Markov chain structures, which are related to the concept of weak Markov. Markov chain structures are key objects of interest in modeling structured dependence of Markovian type between stochastic dynamical given in terms of Markov chains. Accordingly, much of the discussion presented in this chapter is devoted to construction of Markov chain structures. Our construction allows for accommodating in a Markov structure model various dependence structures exhibited by phenomena one wants to model.

In this chapter we define and study strong Markov family structures for a collection of time-homogeneous nice R-Feller–Markov families. Markov structures are key objects of interest in modeling structured dependence of Markovian type between stochastic dynamical systems of Markovian type, such as Markov families or Markov processes. Much of the discussion presented in this chapter is devoted to construction of Markov structures. Part of the input to any respective construction procedure is provided by marginal data, which we refer to as marginal inputs. Another part of the input is provided by data and/or postulates regarding stochastic dependence between the coordinates of the resulting Markov structure, which we refer to as dependence structure input. These inputs have to be appropriately accounted for in constructions of Markov structures. This, in principle, can be done, since, as discussed in this chapter, one has quite substantial flexibility in constructing Markov structures, which allows for accommodating in a Markov structure model various dependence structures exhibited by phenomena one wants to model.