A functional limit theorem for the profile of random recursive trees

Let X-n(k) be the number of vertices at level k in a random recursive tree with n + 1 vertices. We prove a functional limit theorem for the vector-valued process (X-[nt] (1), ... ,X-[(nt]) (k))(t >= 0), for each k is an element of N. We show that after proper centering and normalization, this process converges weakly to a vector-valued Gaussian process whose components are integrated Brownian motions. This result is deduced from a functional limit theorem for Crump-Mode-Jagers branching processes generated by increasing random walks with increments that have finite second moment. Let Y-k(t) be the number of the kth generation individuals born at times <= t in this process. Then, it is shown that the appropriately centered and normalized vector-valued process (Y-1(St), ... ,Y-k(st))(t >= 0 )converges weakly, as s -> infinity, to the same limiting Gaussian process as above.