Distribution of sum of two independent Poisson random variables $\mathcal{P_1}$, $\mathcal{P_2}$ with dependent rates

Question

Let say I have $\mathcal{P_1} \sim \text{Poisson}(\lambda_1) $ and $\mathcal{P_2} \sim \text{Poisson}(\lambda_2)$, where $\lambda_1$ and $\lambda_2$ are two dependent random variables. I know that if $\lambda_1$ and $\lambda_2$ are indendent random variables then we know that the distribution of $\mathcal{P_1}+\mathcal{P_2}$ is $\text{Poisson}(\lambda_1+\lambda_2)$. However, it is not clear to me what happen if $\lambda_1$ and $\lambda_2$ are dependent rdvs. Is there any result for this kind of problems? Many thanks!