High degrees in random recursive trees

Abstract For , let T n be a random recursive tree (RRT) on the vertex set . Let be the degree of vertex v in T n , that is, the number of children of v in T n . Devroye and Lu showed that the maximum degree Δ n of T n satisfies almost surely; Goh and Schmutz showed distributional convergence of along suitable subsequences. In this work we show how a version of Kingman's coalescent can be used to access much finer properties of the degree distribution in T n . For any , let . Also, let be a Poisson point process on with rate function . We show that, up to lattice effects, the vectors converge weakly in distribution to . We also prove asymptotic normality of when slowly, and obtain precise asymptotics for when and is not too large. Our results recover and extends the previous distributional convergence results on maximal and near‐maximal degrees in RRT.