Let formula_1 be some measure space with formula_2-finite measure formula_3. The Poisson random measure with intensity measure formula_3 is a family of random variables formula_5 defined on some probability space formula_6 such that
i) formula_7 is a Poisson random variable with rate formula_8.
ii) If sets formula_9 don't intersect then the corresponding random variables from i) are mutually independent.
iii) formula_10 is a measure on formula_11
Existence.
If formula_12 then formula_13 satisfies the conditions i)–iii). Otherwise, in the case of finite measure formula_3, given formula_15, a Poisson random variable with rate formula_16, and formula_17, mutually independent random variables with distribution formula_18, define formula_19 where formula_20 is a degenerate measure located in formula_21. Then formula_22 will be a Poisson random measure. In the case formula_3 is not finite the measure formula_22 can be obtained from the measures constructed above on parts of formula_25 where formula_3 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.