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.


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
Full Text: DOI arXiv Euclid


[1] Backhausz, A. and Móri, T. F.: Degree distribution in the lower levels of the uniform recursive tree. Annales Univ. Sci. Budapest., Sect. Comp.36, (2012), 53–62. · Zbl 1249.05355
[2] Billingsley, P.: Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney, 1968. xii+253 pp. · Zbl 0172.21201
[3] Carlsson, H. and Nerman, O.: An alternative proof of Lorden’s renewal inequality. Adv. Appl. Probab.18, (1986), 1015–1016. · Zbl 0606.60082
[4] Chauvin, B., Drmota, M. and Jabbour-Hattab, J.: The profile of binary search trees. Ann. Appl. Probab.11, (2001), 1042–1062. · Zbl 1012.60038
[5] Chauvin, B., Klein, T., Marckert, J.-F. and Rouault, A.: Martingales and profile of binary search trees. Electron. J. Probab.10, (2005), 420–435. · Zbl 1109.60059
[6] Devroye, L.: Branching processes in the analysis of the heights of trees. Acta Inform.24, (1987), 277–298. · Zbl 0643.60065
[7] Drmota, M.: Random trees. An interplay between combinatorics and probability. SpringerWienNewYork, Vienna, 2009. xviii+458 pp. · Zbl 1170.05022
[8] Drmota, M., Janson, S. and Neininger, R.: A functional limit theorem for the profile of search trees. Ann. Appl. Probab.18, (2008), 288–333. · Zbl 1143.68019
[9] Flajolet, Ph. and Sedgewick, R.: Analytic combinatorics. Cambridge University Press, Cambridge, 2009. xiv+810 pp. · Zbl 1165.05001
[10] Fuchs, M., Hwang, H.-K. and Neininger, R.: Profiles of random trees: limit theorems for random recursive trees and binary search trees. Algorithmica. 46, (2006), 367–407. · Zbl 1106.68083
[11] Gut, A.: On the moments and limit distributions of some first passage times. Ann. Probab.2, (1974), 277–308. · Zbl 0278.60031
[12] Gut, A: Stopped random walks. Limit theorems and applications. 2nd Edition, Springer, New York, 2009. xiv+263 pp. · Zbl 1166.60001
[13] Iksanov, A.: Functional limit theorems for renewal shot noise processes with increasing response functions. Stoch. Proc. Appl.123, (2013), 1987–2010. · Zbl 1302.60058
[14] Iksanov, A.: Renewal theory for perturbed random walks and similar processes. Birkhäuser/Springer, Cham, 2016. xiv+250 pp. · Zbl 1382.60004
[15] Iksanov, A., Marynych, A. and Meiners, M.: Moment convergence of first-passage times in renewal theory. Stat. Probab. Letters. 119, (2016), 134–143. · Zbl 1350.60091
[16] Iksanov, A., Marynych, A. and Meiners, M.: Asymptotics of random processes with immigration I: Scaling limits. Bernoulli. 23, (2017), 1233–1278. · Zbl 1386.60122
[17] Jabbour-Hattab, J.: Martingales and large deviations for binary search trees. Random Struct. Algor.19, (2001), 112–127. · Zbl 0983.60034
[18] Kabluchko, Z., Marynych, A. and Sulzbach, H.: General Edgeworth expansions with applications to profiles of random trees. Ann. Appl. Probab.27, (2017), 3478–3524. · Zbl 1382.60068
[19] Pittel, B.: Note on the heights of random recursive trees and random \(m\)-ary search trees. Random Struct. Algor.5, (1994), 337–347. · Zbl 0790.05077
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