A functional interpretation of cycloid processes: Application to the study of asymptotic behaviour (Q1969261)

Given a finite set \(S\) and an irreducible homogeneous \(S\)-state Markov chain \(\xi= \{\xi_n\}\), the authors show that there exists a unique collection of elementary cycloids \(\{\gamma_1,\dots, \gamma_B\}\) and strictly positive numbers \(w_1,\dots, w_B\), for which a certain set of decomposition equations holds. The collection \(\{\{y_k\}, \{w_k\}\}\), called cycloid representation of \(\xi\), completely determines a Markov process \(\{\xi_n\}\), called a cycloid process, admits an invariant probability distribution, and decomposes its distribution \(\text{Prob} (\xi_n=i, \xi_{n+1}=j)\), \(i,j\in S\), \(n= 0,1,\dots\), into a linear expression. The decomposition of the cycloid representations is explained by its intrinsic homologic nature and implies a formal analogy to the Chapman-Kolmogorov equations of certain Markovian transition semigroups on finite state spaces (when Betti-type elements are identified through an isomorphism). The obtained decomposition relations of the cycloid representations are called cycloid Chapman-Kolmogorov-type equations. These equations are of particular interest when the coding problem occurring in dynamical systems is formulated in terms of Markov chains. The interpretation of the relationship between the classical Chapman-Kolmogorov equations and the cycloid Chapman-Kolmogorov-type equations is proved to be useful to the study of the asymptotic behaviour of cycloid processes.