Counterexample to a transition probability formula for the ancestral process

Abstract: We consider weighted particle systems in which new generations are re-sampled from current particles with probabilities proportional to their weights. This covers a broad class of sequential Monte Carlo methods, widely used in applied statistics. We consider the genealogical tree embedded into such particle system. When the time is reversed, the particle system induces a partition valued family of processes (partitions on the leaves of the genealogical tree). Our aim here is to give a counterexample to a well known formula describing the transition probabilities of this process.