The Kingman tree length process has infinite quadratic variation

Abstract

In the case of neutral populations of fixed sizes in equilibrium whose genealogies are described by the Kingman N-coalescent back from time t consider the associated processes of total tree length as t increases. We show that the (càdlàg) process to which the sequence of compensated tree length processes converges as N tends to infinity is a process of infinite quadratic variation; therefore this process cannot be a semimartingale. This answers a question posed in Pfaffelhuber et al. (2011).