Imprecise Markov chains with absorption
The authors consider convergence of Markov chains with uncertain parameters, known as imprecise Markov chains, which contain an absorbing state. The paper is structured in 6 sections. After Introduction, in Section 2 are given the most necessary conditions of the model of imprecise Markov chains used. For a Markov chain \(\chi= \{X(n), n= 0,\dots\}\), where \(X(n)\) represents the value (equivalently, the state) of the process at time \(n\), the set of states is denoted as follows: \[ S=:\{-1,0,\dots,s\}:= \{-1\}\cup C, \] where \(-1\) is an absorbing state, and \(C\) is a finite set of transient states. Let \(P\) be a transition matrix. All potential transition matrices for a given time step belong, by definition, to the set \[ {\mathcal P}:= \{P|{\mathbf p}^{(i)}\in{\mathcal P}^{(i)},\,\forall i\in S\}= \{1,0,\dots, 0\}\times \prod_{i\in C}{\mathcal P}^{(i)}, \] where \({\mathcal P}^{(i)}\) are closed sets of probabilities. Among other conditions it is supposed that for all \(P\in{\mathcal P}\), \(C\) is a single aperiodic communicating class. Any set of initial distributions, denoted in general by \({\mathcal D}_0\), will have the property that \({\mathcal D}_0\subseteq{\mathcal M}_0\), where \({\mathcal M}_0\) is the notation for the set of all possible probability mass functions. Then, methods for calculating sets of distributions at general steps are given in Section 3. In the next section is presented, for the imprecise case, a method for conditioning on non-absorption. Section 5 deals with convergence to equilibrium. The authors prove that as time \(n\) approaches infinity the set \({\mathcal D}^C_n\) will tend to \({\mathcal M}^C_\infty\) independently of the set \({\mathcal D}^C_0\subseteq{\mathcal M}_0^C\) chosen as the set of initial distributions; it is not necessary to begin with \({\mathcal M}^C_0\). They emphasize that this process requires a distance function between probability distributions, and based on this a distance between sets of probabilities, that will be used to measure the convergence. Two examples are presented in Section 6. In the last section, ``Concluding remarks'', is proved that the long-term behaviour of such chains is uninteresting, and so adapted the results to allow for conditioning on non-absorption. Some technical results and most of the proofs are included into Appendix A. It is a good and instructive study.
scientific article; zbMATH DE number 5843185
scientific article