Correlation between poisoon processes

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Two Poisson processes cannot be negatively correlated due to the inherent positivity of Poisson random variables, which ensures that their joint distribution remains non-negative. The Frechet bounds can be applied to demonstrate this relationship, highlighting that the correlation between two independent Poisson processes cannot fall below zero. While some discussions suggest the possibility of constructing a negatively correlated Poisson distribution, this contradicts established results from copula theory. A relevant survey article by Embrechts provides insights into these concepts, although it indicates that the initial conjecture about negative correlation may not hold true. The conversation underscores the importance of understanding the mathematical foundations behind Poisson processes and their correlations.
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How to prove that two poisson processes can never be negatively correlated.
 
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Use the Frechet bounds and positivity of Poisson random variables.
 
Thanks bpet. But what is a frechet bound?
I have probability/math background.No physics back ground.
 
The result is from copula theory. A good survey article is "Copulas: a personal view" by Embrechts which has a result relevant to your conjecture (which you'll find is not quite true).
 


ok.yes it may be false actually.poisson distribution with negative correlation can be constructed.from there to poisson process may not be a big leap.Thanks very much for pointing to the article.
 

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