Sum of correlated Poisson random variables mean and variance?

#1
Usually, we can use the following technique use to independent case, If we add m number of identity random variables (like the name as sum(X)=Y) then we can write E(Y)=m*E(X) and Var(Y)=m*Var(X). But dependent (correlated) case we cannot use this way. I am working Poisson, Poisson gamma and Poisson lognormal distributions so, anybody has any suggestions?