Zusammenfassung
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
stochastic process has been studied previously when the population has a
constant large size and evolves via the diffusion limit of standard
Wright-Fisher dynamics. We investigate cases in which the population varies in
size and evolves according to a class of models that includes suitably
conditioned $(1+\beta)$-stable continuous state branching processes (in
particular, it includes the conditioned Feller continuous state branching
process). We also consider the discrete time Markov chain that tracks the MRCA
age 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. We also introduce a new family of Markov
processes that stand in a relation with respect to the $(1+\beta)$-stable
continuous state branching process that is similar to the one between the
Bessel-squared diffusions and the Feller continuous state branching process.
Nutzer