be some measure space with
The Poisson random measure with intensity measure
is a family of random variables
defined on some probability space
is a Poisson random variable with rate
μ (
ii) If sets
don't intersect then the corresponding random variables from i) are mutually independent.
μ ≡ 0
satisfies the conditions i)–iii).
Otherwise, in the case of finite measure
μ
, a Poisson random variable with rate
μ (
, mutually independent random variables with distribution
μ
μ (
is a degenerate measure located in
will be a Poisson random measure.
is not finite the measure
can be obtained from the measures constructed above on parts of
This kind of random measure is often used when describing jumps of stochastic processes, in particular in Lévy–Itō decomposition of the Lévy processes.
The Poisson random measure generalizes to the Poisson-type random measures, where members of the PT family are invariant under restriction to a subspace.