I think the title might capture the entire question but for clarity's sake let me expand here. I have several Poisson distributed random variables, and want to model the product of pairs of these variables. I have reason to believe that many of these variables will be correlated with each other, so it seems like a Poisson model would not be a reasonable approximation of all of the resulting distributions. Is there a distribution that might be appropriate that doesn't assume independence of the variables I'm taking the product of?
This paper seems to discuss multivariate poisson distribution with covariance structures. Check it out. Happy reading!