Weak convergence of the number of vertices at intermediate levels of random recursive trees

Abstract

Let Xn(k) be the number of vertices at level k in a random recursive tree with n + 1 vertices. We are interested in the asymptotic behavior of Xn(k) for intermediate levels k = kn satisfying kn → ∞ and kn = o(log n) as n → ∞. In particular, we prove weak convergence of finite-dimensional distributions for the process (Xn([knu])) u>0 , properly normalized and centered, as n → ∞. The limit is a centered Gaussian process with covariance (u, v) → (u + v) −1 . One-dimensional distributional convergence of Xn(kn), properly normalized and centered, was obtained with the help of analytic tools by Fuchs, Hwang and Neininger in [5]. In contrast, our proofs which are probabilistic in nature exploit a connection of our model with certain Crump-Mode-Jagers branching processes.

Limit theorems for discounted convergent perpetuities

“…Such a process has appeared in the recent articles [8], [17] and [18]. The latter paper provides additional references.…”

Limit theorems for discounted convergent perpetuities

“…In view of (17), this is the general term of a convergent series. Hence, the second series on the right-hand side of ( 18) converges.…”

Weak convergence of random processes with immigration at random times

“…We refer to [6] and [10] for detailed surveys concerning earlier works on random processes with immigration at the epochs of a Poisson or renewal process. A non-exhaustive list of more recent contributions, not covered in the cited sources, includes [7], [8], [9], [12] and [13].…”