Poisson random measure
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Let 
be some measurable space with σ-finite measure μ. The Poisson random measure with intensity measure μ is a family of random variables 
defined on some probability space 
such that
i) 
is a Poisson random variable with rate μ(A).
ii) If sets 
don't intersect then the corresponding random variables from i) are mutually independent.
iii) 
is a measure on 
[edit] Existence
If 
then 
satisfies the conditions i)-iii). Otherwise, in the case of finite measure μ given Z - Poisson random variable with rate μ(E) and 
- mutually independent random variables with distribution 
define 
where δc(A) is a degenerate measure located in c. Then N will be a Poisson random measure. In the case μ is not finite the measure N can be obtained from the measures constructed above on parts of E where μ is finite.
[edit] Applications
This kind of random measures ist often used when describing jumps of stochastic processes, in particular in Lévy-Itō decomposition of the Lévy processes.
[edit] References
- Sato K. Lévy Processes and Infinitely Divisible Distributions Cambridge University Press, (1st ed.) ISBN 0-521-55302-4.
