A binomial process is a special point process in probability theory.

Definition

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Let   be a probability distribution and   be a fixed natural number. Let   be i.i.d. random variables with distribution  , so   for all  .

Then the binomial process based on n and P is the random measure

 

where  

Properties

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Name

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The name of a binomial process is derived from the fact that for all measurable sets   the random variable   follows a binomial distribution with parameters   and  :

 

Laplace-transform

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The Laplace transform of a binomial process is given by

 

for all positive measurable functions  .

Intensity measure

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The intensity measure   of a binomial process   is given by

 

Generalizations

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A generalization of binomial processes are mixed binomial processes. In these point processes, the number of points is not deterministic like it is with binomial processes, but is determined by a random variable  . Therefore mixed binomial processes conditioned on   are binomial process based on   and  .

Literature

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  • Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.