何辉教授,北京师范大学

We consider the genealogical tree of a stationary continuous state branching process with immigration. For a sub-critical stable branching mechanism, we consider the genealogical tree of the extant population at some fixed time and prove that, up to a deterministic time-change, it is distributed as a continuous-time Galton-Watson process with immigration. As a by-product, we study the sizes of the families of the extant population ranked according to their immigration time. The vector of the sizes of these families in the stable case is distributed as the jumps of a time-changed subordinator which yields a Poisson-Kingman distribution. In the quadratic case this corresponds to a Poisson-Dirichlet distribution.