Electronic Communications in Probability

A functional limit theorem for the profile of random recursive trees

Alexander Iksanov and Zakhar Kabluchko

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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.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 87, 13 pp.

Received: 14 January 2018
Accepted: 4 November 2018
First available in Project Euclid: 23 November 2018

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Zentralblatt MATH identifier

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

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

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Iksanov, Alexander; Kabluchko, Zakhar. A functional limit theorem for the profile of random recursive trees. Electron. Commun. Probab. 23 (2018), paper no. 87, 13 pp. doi:10.1214/18-ECP188. https://projecteuclid.org/euclid.ecp/1542942176

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  • [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.
  • [2] Billingsley, P.: Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney, 1968. xii+253 pp.
  • [3] Carlsson, H. and Nerman, O.: An alternative proof of Lorden’s renewal inequality. Adv. Appl. Probab. 18, (1986), 1015–1016.
  • [4] Chauvin, B., Drmota, M. and Jabbour-Hattab, J.: The profile of binary search trees. Ann. Appl. Probab. 11, (2001), 1042–1062.
  • [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.
  • [6] Devroye, L.: Branching processes in the analysis of the heights of trees. Acta Inform. 24, (1987), 277–298.
  • [7] Drmota, M.: Random trees. An interplay between combinatorics and probability. SpringerWienNewYork, Vienna, 2009. xviii+458 pp.
  • [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.
  • [9] Flajolet, Ph. and Sedgewick, R.: Analytic combinatorics. Cambridge University Press, Cambridge, 2009. xiv+810 pp.
  • [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.
  • [11] Gut, A.: On the moments and limit distributions of some first passage times. Ann. Probab. 2, (1974), 277–308.
  • [12] Gut, A: Stopped random walks. Limit theorems and applications. 2nd Edition, Springer, New York, 2009. xiv+263 pp.
  • [13] Iksanov, A.: Functional limit theorems for renewal shot noise processes with increasing response functions. Stoch. Proc. Appl. 123, (2013), 1987–2010.
  • [14] Iksanov, A.: Renewal theory for perturbed random walks and similar processes. Birkhäuser/Springer, Cham, 2016. xiv+250 pp.
  • [15] Iksanov, A., Marynych, A. and Meiners, M.: Moment convergence of first-passage times in renewal theory. Stat. Probab. Letters. 119, (2016), 134–143.
  • [16] Iksanov, A., Marynych, A. and Meiners, M.: Asymptotics of random processes with immigration I: Scaling limits. Bernoulli. 23, (2017), 1233–1278.
  • [17] Jabbour-Hattab, J.: Martingales and large deviations for binary search trees. Random Struct. Algor. 19, (2001), 112–127.
  • [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.
  • [19] Pittel, B.: Note on the heights of random recursive trees and random $m$-ary search trees. Random Struct. Algor. 5, (1994), 337–347.