Random measure

In probability theory, a random measure is a measure-valued random element.^[1][2] Let X be a complete separable metric space and $\mathfrak{B}(X)$ the σ-algebra of its Borel sets. A Borel measure μ on X is boundedly finite if μ(A) < ∞ for every bounded Borel set A. Let $M_X$ be the space of all boundedly finite measures on $\mathfrak{B}(X)$. Let (Ω, ℱ, P) be a probability space, then a random measure maps from this probability space to the measurable space ($M_X$, $\mathfrak{B}(M_X)$).^[3]A measure generally might be decomposed as:

$\mu=\mu_d + \mu_a = \mu_d + \sum_{n=1}^N \kappa_n \delta_{X_n},$

Here $\mu_d$ is a diffuse measure without atoms, while $\mu_a$ is a purely atomic measure.

Random counting measure

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^[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 $X_n$. The diffuse component $\mu_d$ is null for a counting measure.

In the formal notation of above a random counting measure is a map from a probability space to the measurable space ($N_X$, $\mathfrak{B}(N_X)$) a measurable space. Here $N_X$ is the space of all boundedly finite integer-valued measures $N \in M_X$ (called counting measures).

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.^[4]

See also

• Point process
• Poisson random measure
• Random element
• Vector measure
• Ensemble
• Probability measure

References

1. Kallenberg, O., Random Measures, 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin (1986). ISBN 0-12-394960-2 MR 854102. An authoritative but rather difficult reference.
2. Jan Grandell, Point processes and random measures, Advances in Applied Probability 9 (1977) 502-526. MR 0478331 JSTOR A nice and clear introduction.
3. Daley, D. J.; Vere-Jones, D. (2003). "An Introduction to the Theory of Point Processes". Probability and its Applications. doi:10.1007/b97277. ISBN 0-387-95541-0.
4. 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