View source for Poisson random measure

Let <math>(E, \mathcal A, \mu)</math> be some [[measure space]] with <math>\sigma</math>-[[sigma finite measure|finite measure]] <math>\mu</math>. The '''Poisson random measure''' with intensity [[Measure (mathematics)|measure]] <math>\mu</math> is a family of [[random variables]] <math>\{N_A\}_{A\in\mathcal{A}}</math> defined on some [[probability space]] <math>(\Omega, \mathcal F, \mathrm{P})</math> such that

i) <math>\forall A\in\mathcal{A},\quad N_A</math> is a [[Poisson distribution|Poisson random variable]] with rate <math>\mu(A)</math>.

ii) If sets <math>A_1,A_2,\ldots,A_n\in\mathcal{A}</math> don't intersect then the corresponding [[random variables]] from i) are mutually [[Statistical independence|independent]].

iii) <math>\forall\omega\in\Omega\;N_{\bullet}(\omega)</math> is a measure on <math>(E, \mathcal {A})</math>

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

==Applications==
This kind of [[random measure]] is often used when describing jumps of [[stochastic process]]es, in particular in [[Lévy–Itō decomposition]] of the [[Lévy process]]es.

==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==
* {{cite book |last=Sato |first=K. |year=2010 |title=Lévy Processes and Infinitely Divisible Distributions |publisher=Cambridge University Press |isbn=978-0-521-55302-5 }}

[[Category:Statistical randomness]]
[[Category:Poisson point processes]]