Summary
If a finite sequence of independent (not necessarily stationary) renewal processes is given, a superposition process can easily be defined as the union of all point sequences represented by the given processes. The properties of such superposition processes are investigated. First, a necessary and sufficient condition for a superposition process to be a renewal process is given. Essentially, this condition reads thus: the given processes must be Poisson processes. The main result given in this paper is a limit theorem for superposition processes which shows that, even with largely arbitrary renewal processes superimposed, the superposition process has local properties which approach the properties of the Poisson process as the number of given processes increases. The theorem contains some well-known special theorems of this type [e.g. Khintchine, 1960; Franken, 1963].
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Von der Fakultät für Allgemeine Wissenschaften der T. H. München angenommene Habilitationsschrift (Auszug).
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Störmer, H. Zur Überlagerung von Erneuerungsprozessen. Z. Wahrscheinlichkeitstheorie verw Gebiete 13, 9–24 (1969). https://doi.org/10.1007/BF00535794
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DOI: https://doi.org/10.1007/BF00535794