We show that independent superposition of Markov point processes with respect to the same neighbourhood relation preserves the Hammersley--Clifford factorisation up to second order. If the processes are identically distributed, the third order interaction structure is preserved as well. Finally, we prove that the superposition of standardised locally stable Markov point processes converges weakly to a Poisson process.

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van Lieshout, M.N.M. (1999). A note on the superposition of Markov point processes. CWI. Probability, Networks and Algorithms [PNA]. CWI.