Random measure

In probability theory, a random measure is a measure-valued random element.Kallenberg, O., "Random Measures", 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin (1986). ISBN 0-123-94960-2 [http://www.ams.org/mathscinet-getitem?mr=854102 MR854102] . An authoritative but rather difficult reference.

] Jan Grandell, Point processes and random measures, "Advances in Applied Probability" 9 (1977) 502-526. [http://www.ams.org/mathscinet-getitem?mr=0478331 MR0478331] [http://links.jstor.org/sici?sici=0001-8678%28197709%299%3A3%3C502%3APPARM%3E2.0.CO%3B2-5 JSTOR] A nice and clear introduction.

] A random measure of the form

:$mu=sum\_\{n=1\}^N\; delta\_\{X\_n\},$

where $delta$ is the Dirac measure, and $X\_n$ 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 $X\_n$. Random measures are useful in the description and analysis of Monte Carlo methods, such as Monte Carlo numerical quadrature and particle filters [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] .

ee also

* Point process
* Poisson random measure
* Random element
* Vector measure
* Ensemble

References