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By Thomas J. Enright

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Specialization to the ﬁnite state space case Let us consider the special case of a ﬁnite state space semi-Markov chain. So, the state space is E = {1, . . , s}, s < ∞. We want to see how the results on Markov renewal theory obtained in the inﬁnite case are modiﬁed here and also to present an example of a four-state semi-Markov system. 1. Markov renewal equation in the ﬁnite state space case. The objective of the next proposition is to give a necessary and suﬃcient condition for the existence and uniqueness of the left inverse.

D, and Bl = {β ∈ {−1, 1}l , β1 = 1}. A Fourier lattice design in d dimensions, with a single d-dimensional generator g = (g1 , . . , n n , j = 0, . . , n − 1 . We may take g1 = 1, when g1 and n are mutually prime. See [7] for the study of such designs in relation to low-discrepancy problems in number theory and integration. In the same way that an equally spaced design on [0, 1] is D optimum for the one dimensional Fourier model of order m, provided the sample size n ≥ 2m + 1, Fourier lattice designs are D-optimal for Fourier models for special choices of g and n.

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