Asymptotic size of families in some generalized Fleming-Viot models
A talk in the Other series by
Grégoire Véchambre from AMSS, Chinese Academy of Sciences
| Abstract: | Generalized Fleming-Viot processes, also called flows of bridges, model the evolution of a constant size population represented by the interval $[0,1]$. In a series of papers, Bertoin and Le Gall have shown that those processes were in one-to-one correspondence with exchangeable coalescents and studied the inverse flows. In this work in progress, we consider a population at time $0$ that has been subject to some generalized Fleming-Viot dynamic at all times in the past. We define families of level t as sub-populations from time 0 that share the same ancestor at time $-t$ in the past and we denote by $W_k(t)$ the size of the $k^{th}$ largest family of level $t$. Our focus is mainly on the asymptotic behavior of the expected sizes of the families. In particular, we show that all $E[W_k(t)]$ for $k \geq 2$ decay exponentially as $t$ goes to infinity and provide exact expressions for their decay rates. It turns out that, under our assumptions, there is a critical number $N \geq 2$, depending on the parameter of the model, such that the decay rates of $E[W_k(t)]$ are the same for all $k \geq N$ (this does not violate the requirement $\sum_{k\geq 1} W_k(t) \leq 1$). We will give some ideas of proof and propose interpretations of this phenomenon. Within the CRC this talk is associated to the project(s): C1 |