Small-time Sampling Behaviour of a Fleming-Viot Process

Abstract

The Fleming-Viot process with parent-independent mutation process is one particular neutral population genetic model. As time goes by, some initial species are replaced by mutated ones gradually. Once the population mutation rate is high, mutated species will elbow out all the initial species very quickly. Small time behaviour in this case seems to be the key to understand this fast transition. The small-time asymptotic results related to time scale $\frac{t}{\theta}$ and $a(\theta)t$, where $\lim_{\theta\to\infty}\theta a(\theta)=0$, are obtained in \cite{MR1815182},\cite{MR1649005}, \cite{MR1887170} and \cite{MR2184086}, respectively. Only the behaviour under the scale $t(\theta)$, where $\lim_{\theta\to\infty}\theta a(\theta)=\infty$, was left untouched. In this paper, the weak limits under various small time scales are obtained. Of particular interest is the large deviations for the small-time transient sampling distributions, which reveal interesting phase transition. Interestingly, such a phase transition is uniquely determined by some species diversity indices.