The multivariate Poisson distribution MultiPoissonDistribution[0, ] with ={1, 2, ...} is the distribution followed by a Poisson with mean 0 summed with a vector of independent Poissons with means 1, 2,....
The parameter 0 and the elements of the vector can be any positive numbers.
MultiPoissonDistribution can be used with such functions as Mean, PDF and RandomInteger.