Question
Choose the option which best completes the following statement:
In continuous time,
$(1)\quad$ for an irreducible birth and death process to have a steady state distribution, its embedded chain has to be periodic.
$(2)\quad$ any inhomogenous Poisson process can be represented by applying a suitable time transformation to a homogenous Poisson process.
$(3)\quad$ any pure death process can be represented by applying a suitable state space transformation to a homogenous Poisson process.
$(4)\quad$ any pure birth process can be represented by applying a suitable time transformation to a homogenous Poisson process.
My thoughts
This appeared in a practice paper my professor gave to us in preparation for our final examination. When I first did this, I chose $(2)$, as I believe it was what I had learnt in class. However, the solution is apparently $(4)$.
I discussed this with my professor and he, too, could not see why $(2)$ is incorrect, or rather, given how the question is phrased, why would $(4)$ be more accurate than $(2)$?
He did say that he did not set this question, so it might very well be possible that the question is flawed/incorrectly phrased, but assuming this is not the case, I was hoping to get some insight from the community here.