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Finite non-homogeneous semi-Markov processes: Theoretical and computational aspects. (English) Zbl 0546.60087
The authors are concerned with finite state semi-Markov processes in a time-nonhomogeneous setting. Many definitions are given but only few results are proved. (Note that in definition 3 the use of the adjective ”multiple” for the imbedded Markov chain is simply wrong.) Special attention is given to the discrete-time case. An algorithm is thus suggested for obtaining the transition probabilities \(P_{ij}(s,t)\) of a discrete-time process, which in the general case satisfy a system of equations of the form \[ (*)\quad P_{ij}(s,t)= \delta_{ij}(1-H_ i(s,t)+ \sum_{k\in I} \int^{t}_{s}P_{kj}(u,t)Q_{ik}(s,du),\quad i,j\in I. \] The numerical results of the application of the algorithm to a social pension problem are presented in four tables.
Reviewer’s remarks. The authors’ assertion (see p. 157) that the concept of time-homogeneity was never criticized until the paper of the second author in Riv. Mat. Sci. Econ. Soc. 2, 157-167 (1979; Zbl 0492.60088) asks for serious amendments. Actually, the most important part (sections 3 and 4) of that paper are an almost verbatim Italian translation of the paper of A. Iosifescu-Manu, Stud. Cerc. Mat. 24, 529-533 (1972; Zbl 0245.60067). In particular system (*) occurs in the latter paper (see eq. (1), p. 531), where the existence of a minimal solution of it is proved.
\(\{\) Editorial remark: The reviewer added a list of papers where also a considerable overlap of de Dominicis’ publications with other work has to be stated: (1) See the MR review 81i:60081; (2) The paper of Gh. Popescu, Functional limit theorems for random systems with complete connections, Proc. 5th Conf. Probab. Theory, Braşov 1974, 261-275 (1977; Zbl 0373.60031) and the second author’s paper ”Weak convergence of empirical distribution functions for infinite order chains” in Rend. Mat. Appl., VII. Ser. 2, 359-370 (1982); (3) The paper of Gh. Popescu, Asymptotic behaviour for random systems with complete connections, II, Stud. Cerc. Mat. 30, 181-215 (1978; Zbl 0403.60059) and the second author’s papers ”Normalità asintotica di catene Markoviane non \(\phi\)- mixing” in Ill. J. Math. 25, 455-463 (1981; Zbl 0485.60025) and ”Un principio di invarianza per catene Markoviane di elementi aleatori”, in Collect. Math. 31, 125-139 (1980; Zbl 0485.60026); (4) The paper of T. Kaijser, Another central limit theorem for random systems with complete connections, Rev. Roum. Math. Pures Appl. 24, 383-412 (1979; Zbl 0408.60068) and the second author’s paper ”Some limit theorems for Markov chains not necessarily fulfilling Doeblin’s condition”, Ric. Mat. 29, 125-152 (1980; Zbl 0471.60076).\(\}\)
Reviewer: M.Iosifescu

60K15 Markov renewal processes, semi-Markov processes
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