The Moran Genealogy Process
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We give a novel representation of the Moran Genealogy Process, a continuous-time Markov process on the space of size-n genealogies with the demography of the classical Moran process. We derive the generator and unique stationary distribution of the process and establish its uniform ergodicity. In particular, we show that any initial distribution converges exponentially to the probability measure identical to that of the Kingman coalescent. We go on to show that one-time sampling projects this stationary distribution onto a smaller-size version of itself. Next, we extend the Moran genealogy process to include sampling through time. This allows us to define the Sampled Moran Genealogy Process, another Markov process on the space of genealogies. We derive exact conditional and unconditional probability distributions for this process under the assumption of stationarity, and an expression for the likelihood of any sequence of genealogies it generates. This leads to some interesting observations pertinent to existing phylodynamic methods in the literature.
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