Let $\{N_{1}(t)\}_{t\geq0}$ and $\{N_{2}(t)\}_{t\geq 0}$ be two independent birth processes with strictly positive birth rates $\lambda_{n}^{(1)}$ and $\lambda_{n}^{(2)},n\geq 0$ respectively.
Define $X(t)=N_{1}(t)+N_{2}(t)$ and $Y(t)=N_{1}(t)-N_{2}(t)$.
The exercise given for this situation asks me to identify which of 4 statements is true. The statement that apparently is true is
"The process $\{N_{1}(t),X(t)\}_{t\geq 0}$ is a Markov process."
Could someone help me in showing how this is a Markov process? I am confused on how to combine $N_{1}(t)$ and $X(t)$.
The false statements were:
- "The process $\{N_{2}(t),Y(t)\}_{t\geq 0}$ is not a Markov process."
- "The process $\{X(t)\}_{t\geq 0}$ is a birth process."
- "The process $\{Y(t)\}_{t\geq 0}$ may be a birth-death process."