Rescaled stable generalised Fleming-Viot processes : Flickering random measures

Abstract

We show how Donnelly and Kurtz' (modified) lookdown construction for measure-valued processes can be used to analyse the longterm- and scaling properties of spatially stable generalised $Lambda$-Fleming Viot processes, exhibiting a rare ``natural'' example of a scaling family converging in f.d.d. sense, but not in any of Skorohod's topologies on path space. This completes results of Fleischmann and Wachtel (2004) about the spatial Neveu process and complements results of Dawson and Hochberg (1982) about the classical Fleming Viot process. The lookdown construction provides an elegant machinery and clear intuition to describe the path properties of the family in terms of a ``flicker effect'', clarifying ``what can go wrong.''