Coalescent process point process of a branching process in varying environment

Abstract

Consider a doubly infinite branching tree in varying environment. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process (A i , i ≥ 1), where A i is the coalescent time between individuals i and i + 1. In general, this process is not Markovian. In constant environment, Lambert and Popovic (2013) proposed a Markov process of point measures (B i , i ≥ 1) to reconstruct the coalescent point process. We present a counterexample where we show that their process has not the Markov property.We define a vector valued Markov process with the minimal amount of information to reconstruct the genealogy of the standing population. We highlight that this process is the Markovian correction of (B i , i ≥ 1). Finally, when the offspring distributions are lineal fractional, we show that the variables (A i , i ≥ 1) are independent and identically distributed.