  If we follow an asexually reproducing population through time, then the
amount of time that has passed since the most recent common ancestor (MRCA) of
all current individuals lived will change as time progresses. The resulting
"MRCA age" process has been studied previously when the population has a
constant large size and evolves via the diffusion limit of standard
Wright--Fisher dynamics. For any population model, the sample paths of the MRCA
age process are made up of periods of linear upward drift with slope +1
punctuated by downward jumps. We build other Markov processes that have such
paths from Poisson point processes on $\mathbb{R}_{++}\times\mathbb{R}_{++}$
with intensity measures of the form $\lambda\otimes\mu$ where $\lambda$ is
Lebesgue measure, and $\mu$ (the "family lifetime measure") is an arbitrary,
absolutely continuous measure satisfying $\mu((0,\infty))=\infty$ and
$\mu((x,\infty))<\infty$ for all $x>0$. Special cases of this construction
describe the time evolution of the MRCA age in $(1+\beta)$-stable continuous
state branching processes conditioned on nonextinction--a particular case of
which, $\beta=1$, is Feller's continuous state branching process conditioned on
nonextinction. As well as the continuous time process, we also consider the
discrete time Markov chain that records the value of the continuous process
just before and after its successive jumps. We find transition probabilities
for both the continuous and discrete time processes, determine when these
processes are transient and recurrent and compute stationary distributions when
they exist.
