Poisson random measure

If \mu\equiv 0 then N\equiv 0 satisfies the conditions i)–iii). Otherwise, in the case of finite measure \mu, given Z, a Poisson random variable with rate \mu(E), and X_{1}, X_{2},\ldots, mutually independent random variables with distribution \frac{\mu}{\mu(E)}, define N_{\cdot}(\omega) = \sum\limits_{i=1}^{Z(\omega)} \delta_{X_i(\omega)}(\cdot) where \delta_{c}(A) is a degenerate measure located in c. Then N will be a Poisson random measure. In the case \mu is not finite the measure N can be obtained from the measures constructed above on parts of E where \mu is finite. ==Applications==

This kind of random measure is often used when describing jumps of stochastic processes, in particular in Lévy–Itō decomposition of the Lévy processes. ==Generalizations==

The Poisson random measure generalizes to the Poisson-type random measures, where members of the PT family are invariant under restriction to a subspace. ==References==