On the genealogy of conditioned stable Levy forests

A Levy forest of size s > 0 is a Poisson pout process in the set of Levy 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 Levy 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 +infinity, 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 Levy process with no negative jumps respectively