## A functional limit theorem for the profile of random recursive trees.(English)Zbl 1406.60051

Summary: 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 $$(X_{[n^t]}(1),\dots , X_{[n^t]}(k))_{t\geq 0}$$, 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 $$(Y_{1}(st),\dots , Y_k(st))_{t\geq 0}$$ converges weakly, as $$s\rightarrow \infty$$, to the same limiting Gaussian process as above.

### MSC:

 60F17 Functional limit theorems; invariance principles 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60G50 Sums of independent random variables; random walks 60C05 Combinatorial probability 60F05 Central limit and other weak theorems
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