Random measure

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In probability theory, a random measure is a measure-valued random element.^[1]^[2] A random measure of the form

$\mu=\sum_{n=1}^N \delta_{X_n},$

where $δ$ is the Dirac measure, and $X n$ are random variables, is called a point process^[1]^[2] 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^[3].

[edit] See also

[edit] References

^ ^a ^b Kallenberg, O., Random Measures, 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin (1986). ISBN 0-123-94960-2 MR854102. An authoritative but rather difficult reference.
^ ^a ^b 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