be some measure space-finite measure. The Poisson random measure with intensity measure is a family of random variables defined on some probability space such that

is a Poisson random variable.

don't intersect then the corresponding random variables from i) are mutually independent.

Existence

satisfies the conditions i)–iii). Otherwise, in the case of finite measure, a Poisson random variable, mutually independent random variables with distribution is a degenerate measure is not finite the measure 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