A functional limit theorem for the profile of random recursive trees

Electronic Communications in Probability

Alexander Iksanov¹ 

关键词: branching random walk;    Crump-Mode-Jagers branching process;    functional limit theorem;    integrated Brownian motion;    low levels;    profile;    random recursive tree;   

DOI  :  10.1214/18-ECP188

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 摘 要 】

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.

【 授权许可】

CC BY