Let $X(t)$ be a birth-death process. If $S_i$ is the sojourn time in state $i$, which cannot be assumed to be exponential, prove that $P(S_i \ge t + h) = P(S_i \ge t) P (S_i \ge h)$.
Book says it follows by the Markov property, but I can't figure out how to show it from that.
I know the Markov property says $P(X(t_k) = i_k | X(t_{k-1}) = i_{k-1}, X(t_{k-2}) = i_{k-2}, \dots) = P(X(t_k) = i_k | X(t_{k-1}) = i_{k-1})$ but I can't figure out the relationship that shows the above equality.
Anyone have any ideas?
Hint: $$\mathbb P\left[S_i > t+h\right] = \mathbb P\left[\bigcap_{0<s\le t+h}\{ X(s) = i\}\Big | X(0) = i\right]=\mathbb P\left[\bigcap_{t<s\le t+h}\{ X(s) = i\}\Big | \bigcap_{0\le s\le t}\{X(s)=i\}\right]\mathbb P\left[\bigcap_{0<s\le t}\{ X(s) = i\}\Big |X(0) = i\right]$$
Using the Markov proprety, $$\mathbb P\left[\bigcap_{t<s\le t+h}\{ X(s) = i\}\Big | \bigcap_{0\le s\le t}\{X(s)=i\}\right] = \mathbb P\left[\bigcap_{t<s\le t+h}\{ X(s) = i\}\Big | X(t) = i\right]$$