A pure-birth Process is a generalization of a homogeneous Poission process. Whereas in the Poisson process the holding times between jumps are iid exponentially distributed random variables with parameter $\lambda$, in a pure-birth process the parameters may vary between jumps, i.e. the waiting between states $n$ and $n+1$ is exponentially distributed with parameter $\lambda_n$.
Let $X_1$, $X_1$ be two independent pure-birth processes with rates $\lambda^1_i, \lambda^2_i$ respectivley. By pairs of $X_1$ and $X_2$ I mean the process $$X(t) := (X_1(t), X_2(t))$$
I'd like to prove that $X$ is a contininuous-time Markov process that starts in state $(0,0)$ and may jump from state $(n,m)$ to either $(n+1,m)$ or $(n,m+1)$ with rates $\lambda^1_n$ and $\lambda^2_m$ respectively, depending on which process $X_1$,$X_2$ the next jumps occurs.
While this might seem very intuitive I'm having trouble relating the properties of $X_1$ and $X_2$ to $X$.
Can you point to a treatment I can apply my problem as a special case of?