Suppose that $\{N_1(t)\,t \ge 0\}$ and $\{N_1(t)\,t \ge 0\}$ are two independent Poisson processes with rates of $\lambda_1$ and $\lambda_2$ respectively, then prove the sum process where $N(t) = N_1(t)+N_2(t)$ for all $t \ge 0$ is a Poisson process with rate of $\lambda_1+\lambda_2$.
The proof on my textbook includes the step of proving that $N(t)$ has independent increments, of which the reason is that $N_1(t)$ and $N_2(t)$ are independent and both possess stationary and independent increments properties. But I couldn't understand such an obvious step.
Let $I_1$ and $I_2$ be two disjoint intervals, then our goal is to show that $N_1(I_1)+N_2(I_1)$ and $N_1(I_1)+N_2(I_1)$ are independent. Obviously, we have:
- $N_1(I_1)$, $N_1(I_2)$, $N_2(I_1)$, $N_2(I_1)$ are independent of each other;
- $(N_1(I_1),N_1(I_2))$ and $(N_2(I_1),N_2(I_2))$ are independent random vectors since $\{N_1(t)\,t \ge 0\}$ and $\{N_1(t)\,t \ge 0\}$ are two independent processes.
Next, I only need to prove that $(N_1(I_1),N_2(I_1))$ and $(N_1(I_2),N_2(I_2))$ are two independent vectors to achieve our goal. But I was stuck here. It seems that I'm complicating the question. Does someone have a better idea? Please help me out here.