Coalescent point process of branching trees in varying environment

Abstract: Consider an arbitrary large population at the present time, originated at an unspecified arbitrary large time in the past, where individuals in the same generation reproduce independently, forward in time, with the same offspring distribution but potentially changing among generations. In other words, the reproduction is driven by a Galton-Watson process in a varying environment. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process $(A_i, i\geq 1)$, where $A_i$ is the coalescent time between individuals $i$ and $i+1$. In general, this process is not Markov. In constant environment, Lambert and Popovic (2013) proposed a Markov process of point measures to reconstruct the coalescent point process. We present a counterexample where we show that their process does not have the Markov property. The main contribution of this work is to propose a vector valued Markov process $(B_i,i\geq 1)$, that reach the goal to reconstruct the genealogy, with finite information for every $i$. Additionally, when the offspring distributions are lineal fractional, we show that the variables $(A_i, i\geq 1)$ are independent and identically distributed.