Let $(X_t)_{t \geq 0}$ be a simple birth process with rates $\lambda_n$, $n\geq 0$ starting from $k$.
The Markov property states that the two processes $(X_t)_{0 \leq t \leq r}$ and $(X_{s+r})_{r \geq 0}$ are conditionally independent given $X_r = k^*$; and that $(X_{s+r})_{r \geq 0}$ is a simple birth process starting from $k^*$.
How do you deduce from this the Markov property for simple birth process at a random positive time $R$, given the density $f_R$ of $R$?
What exactly do I have to show?