RANDOM MEASURE

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 $delta$ is the Kronecker's delta, and $X\_n$ are random variables, is called a ''point process''. 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].

Contents

See also

References

See also

★ Point process

★ Poisson random measure

★ Random element

★ Vector measure

★ Ensemble

References

1. 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.

2. Jan Grandell, Point processes and random measures, ''Advances in Applied Probability'' 9 (1977) 502-526. MR0478331 JSTOR A nice and clear introduction.

3. 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