A functional limit theorem for the profile of random recursive trees

Alexander Iksanov,Zakhar Kabluchko

doi:10.1214/18-ecp188

Abstract

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 $(X_{[n^t]}(1),\ldots , X_{[n^t]}(k))_{t\geq 0}$, for each $k\in \mathbb N$. 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 $k$th generation individuals born at times $\leq t$ in this process. Then, it is shown that the appropriately centered and normalized vector-valued process $(Y_{1}(st),\ldots , Y_k(st))_{t\geq 0}$ converges weakly, as $s\to \infty $, to the same limiting Gaussian process as above.

Highlights

Introduction and main results1.1 Functional limit theorem for random recursive treesAn increasing Cayley tree with n vertices is a rooted tree with vertices labeled with 1, 2 . . . , n that satisfies the following property: the root is labeled with 1, and the labels of the vertices on the unique path from the root to any other vertex form an increasing sequence
We prove a functional limit theorem for the vector-valued process (X[nt](1), . . . , X[nt](k))t≥0, for each k ∈ 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

Summary

Functional limit theorem for random recursive trees

An increasing Cayley tree with n vertices is a rooted tree with vertices labeled with 1, 2 . . . , n that satisfies the following property: the root is labeled with 1, and the labels of the vertices on the unique path from the root to any other vertex (labeled with m ∈ {2, . . . , n}) form an increasing sequence. Apart from [10], we are aware of only one work studying vertices of random recursive trees at a fixed level. It is shown in [1] that the proportion of vertices at level k ∈ N having more than t log n descendants converges to (1 − t)k a.s. a Poisson limit theorem is proved in [1] for the number of vertices at fixed level k that have a fixed number of descendants. The following functional limit theorem holds for the low levels profile of a random recursive tree:. The answer is ‘NO’ because in binary search trees the number of descendants of any node is bounded by 2, which means that these trees become saturated at low levels with probability converging to 1

Functional limit theorem for Crump-Mode-Jagers processes

Strategy of proofs