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 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 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.[3]