where is the Dirac measure, and are random variables, is called a point process or random counting measure. This random measure describes the set of N particles, whose locations are given by the (generally vector valued) random variables . The definitions of expectation measure, Laplace functional, moment measures and stationarity for random measures follow those of point processes. Random measures are useful in the description and analysis of Monte Carlo methods, such as Monte Carlo numerical quadrature and particle filters.
See also 
- Kallenberg, O., Random Measures, 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin (1986). ISBN 0-12-394960-2 MR854102. An authoritative but rather difficult reference.
- Jan Grandell, Point processes and random measures, Advances in Applied Probability 9 (1977) 502-526. MR0478331 JSTOR A nice and clear introduction.
- Crisan, D., Particle Filters: A Theoretical Perspective, in Sequential Monte Carlo in Practice, Doucet, A., de Freitas, N. and Gordon, N. (Eds), Springer, 2001, ISBN 0-387-95146-6
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