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If $N_1$ and $N_2$ are independent Poisson processes then the superposition is a Poisson process. Is it possible to construct two dependent Poisson processes such that the superposition is a Poisson process again? How can I do that? Can I use some copula directly?

An article that suggests this is possible is Pfeifer and Neslehova (2004) - Modelling and generating dependent risk processes for IRM and DPA. How to do it exactly however is unclear to me.

A related question to this topic was asked in: Mixing and dividing point processes

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    Do you mean the marginals are Poisson or the conditionals are Poisson? My Johnson, Kotz, Balakrishnan observes that "There are no bivariate distributions with Poisson marginals for which the conditional distribution of either variable, give the other, is Poisson." There is only one bivariate distribution with Poisson conditionals, and the sum is not Poisson (same source.) Also, the correlation between the two processes must be zero, otherwise the sum can't be Poisson (variance of sum $\ne$ mean of sum). So the nature of the dependence is somewhat restricted... I doubt this is possible. – jbowman Feb 03 '12 at 17:10

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