Mapping theorem (point process)

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

Statement 
Let  be locally compact and polish and let

be a measurable function.  Let  be a Radon measure on  and assume that the pushforward measure

of  under the function  is a Radon measure on . 

Then the following holds: If  is a Poisson point process on  with intensity measure , then  is a Poisson point process on  with intensity measure . 

References 

Category:Poisson point processes