Marked point processes as limits of Markovian arrival streams

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process .N there is a sequence (N"(m)} of such Markovian arrival streams with the property that X(m) . A as m - co. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

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