Coalescent point process of branching trees in a varying environment

Abstract

Consider an arbitrary large population at the present time, originated at an unspecified arbitrary large time in the past, where individuals within the same generation independently reproduce forward in time, sharing a common offspring distribution that may vary across generations. In other words, the reproduction is driven by a Galton-Watson process in a varying environment. The genealogy of the current generation, traced backward in time, is uniquely determined by the coalescent point process $(A_i,i\ge 1)$, where $A_i$ denotes the coalescent time between individuals i and $i+1$. In general, this process lacks the Markov property. In constant environment, Lambert and Popovic (2013) proposed a Markov process of point measures to reconstruct the coalescent point process. We provide a counterexample showing that their process lacks the Markov property. The main contribution of this work is to propose a vector valued Markov process $(B_i,i\ge 1)$, that can reconstruct the genealogy, with finite information for every i. Additionally, in the case of linear fractional offspring distributions, we establish that the variables of the coalescent point process $(A_i,i\ge 1)$ are independent and identically distributed.