On the genealogy of conditioned stable Lévy forests

A Lévy forest of size s > 0 is a Poisson point process in the set of Lévy trees which is defined on the time interval [0, s]. The total mass of this forest is defined by the sum of the masses of all its trees. We give a realization of the stable Lévy forest of a given size conditioned on its total mass using the path of the unconditioned forest. Then, we prove an invariance principle for this conditioned forest by considering k independent Galton-Watson trees whose offspring distribution is in the domain of attraction of any stable law conditioned on their total progeny to be equal to n. We prove that when n and k tend towards +∞, under suitable rescaling, the associated coding random walk, the contour and height processes all converge in law on the Skorokhod space towards the first passage bridge and height process of a stable Lévy process with no negative jumps respectively.