Poisson random forests and coalescents in expanding populations.
|
Mon, 25/02 14:15 |
SAM FINCH (University of Copenhagen) |
Stochastic Analysis Seminar  |
Oxford-Man Institute |
| Let (V, ≥) be a finite, partially ordered set. Say a directed forest on V is a set of directed edges [x,y> with x ≤ y such that no vertex has indegree greater than one. Thus for a finite measure μ on some partially ordered measurable space D we may define a Poisson random forest by choosing a set of vertices V according to a Poisson point process weighted by the number of directed forests on V, and selecting a directed forest uniformly. We give a necessary and sufficient condition for such a process to exist and show that the process may be expressed as a multi-type branching process with type space D. We build on this observation, together with a construction of the simple birth death process due to Kurtz and Rodrigues [2011] to develop a coalescent theory for rapidly expanding populations. | |||