View source for Mapping theorem (point process)

The '''mapping theorem''' is a theorem in the theory of [[point process]]es, a sub-discipline of [[probability theory]]. It describes how a [[Poisson point process]] is altered under [[measurable function|measurable transformations]]. This allows construction of more complex Poisson point processes out of [[homogeneous Poisson point process]]es and can, for example, be used to simulate these more complex Poisson point processes in a similar manner to [[inverse transform sampling]].

== Statement ==
Let <math> X,Y </math> be [[Locally compact space|locally compact]] and [[polish space|polish]] and let
:<math> f \colon X \to Y </math>

be a [[measurable function]]. Let <math> \mu </math> be a [[Radon measure]] on <math> X </math> and assume that the [[pushforward measure]]
:<math> \nu:= \mu \circ f^{-1} </math>

of <math> \mu </math> under the function <math> f </math> is a Radon measure on <math> Y </math>.

Then the following holds: If <math> \xi </math> is a [[Poisson point process]] on <math> X </math> with [[intensity measure]] <math> \mu </math>, then <math> \xi \circ f^{-1} </math> is a Poisson point process on <math> Y </math> with intensity measure <math> \nu:= \mu \circ f^{-1} </math>.<ref name="Klenke531"/>

== References ==
<references>
<ref name="Klenke531">{{cite book |last1=Klenke |first1=Achim |year=2008  |title=Probability Theory |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6|pages=531 }} </ref>
</references>

[[Category:Poisson point processes]]