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In this paper we develop the theory of the so-called W and Z scale matrices for (upwards skip-free) discrete-time and discrete-space Markov additive processes, along the lines of the analogous theory for Markov additive processes in continuous-time. In particular, we provide their probabilistic construction, identify the form of the generating function of W and its connection with the occupation mass formula, which provides the tools for deriving semi-explicit expressions for corresponding exit problems for the upward-skip free process and it's reflections, in terms the scale matrices.

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