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By  and Olav Kallenberg and Olav Kallenberg


Random measure theory is a new and rapidly growing branch of probability of increasing interest both in theory and applications. Loosely speaking, it is concerned with random quantities which can only take non-negative values, such as e.g. the number of random variables in a given sequence possessing a certain property, the time spent by a random process in a certain region etc. In the special case when these quantities are integer valued, our random measures reduce to point processes, which constitute an important subclass of random measures. However, I am convinced that general random measures are potentially just as important from the point of view of applications. (See e.g. Kallenberg (1973c) for some results supporting this opinion.) The justification of random measure theory as a separate topic lies in the overwhelming amount of important results which are specific to random measures in the sense that they do not carryover to general random processes. As a contrast, comparatively few results are specific in this sense to the subclass of point processes. Indeed, it will he clear from the development on the subsequent pages that a substantial part of point process theory, as presented e.g. in the monograph of Kerstan, Matthes and Mecke (1974), carries over to the class of general random measures. The latter may therefore be considered as a natural scope of a theory

Year: 2010
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