Pareto genealogies arising from a Poisson branching evolution model with selection

We study a class of coalescents derived from a sampling procedure out of $N$ i.i.d. Pareto $\left( \alpha \right) $ random variables, normalized by their sum, including $\beta $ –size-biasing on total length effects ( $\beta <\alpha $ ). Depending on the range of $\alpha ,$ we derive the large $N$ limit coalescents structure, leading either to a discrete-time Poisson-Dirichlet $ \left(\alpha ,-\beta \right) \Xi -$ coalescent ( $\alpha \in \left 0,1\right) $ ), or to a family of continuous-time Beta $\left( 2-\alpha ,\alpha -\beta \right) \Lambda -$ coalescents ( $\alpha \in \left 1,2\right) $ ), or to the Kingman coalescent ( $\alpha \ge 2$ ). We indicate that this class of coalescent processes (and their scaling limits) may be viewed as the genealogical processes of some forward in time evolving branching population models including selection effects. In such constant-size population models, the reproduction step, which is based on a fitness-dependent Poisson Point Process with scaling power-law $\left( \alpha \right) $ intensity, is coupled to a selection step consisting of sorting out the $N$ fittest individuals issued from the reproduction step.