## Annals of Applied Probability

### General Edgeworth expansions with applications to profiles of random trees

#### Abstract

We prove an asymptotic Edgeworth expansion for the profiles of certain random trees including binary search trees, random recursive trees and plane-oriented random trees, as the size of the tree goes to infinity. All these models can be seen as special cases of the one-split branching random walk for which we also provide an Edgeworth expansion. These expansions lead to new results on mode, width and occupation numbers of the trees, settling several open problems raised in Devroye and Hwang [Ann. Appl. Probab. 16 (2006) 886–918], Fuchs, Hwang and Neininger [Algorithmica 46 (2006) 367–407], and Drmota and Hwang [Adv. in Appl. Probab. 37 (2005) 321–341]. The aforementioned results are special cases and corollaries of a general theorem: an Edgeworth expansion for an arbitrary sequence of random or deterministic functions $\mathbb{L}_{n}:\mathbb{Z}\to\mathbb{R}$ which converges in the mod-$\phi$-sense. Applications to Stirling numbers of the first kind will be given in a separate paper.

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 6 (2017), 3478-3524.

Dates
First available in Project Euclid: 15 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1513328706

Digital Object Identifier
doi:10.1214/17-AAP1285

Mathematical Reviews number (MathSciNet)
MR3737930

Zentralblatt MATH identifier
1382.60068

#### Citation

Kabluchko, Zakhar; Marynych, Alexander; Sulzbach, Henning. General Edgeworth expansions with applications to profiles of random trees. Ann. Appl. Probab. 27 (2017), no. 6, 3478--3524. doi:10.1214/17-AAP1285. https://projecteuclid.org/euclid.aoap/1513328706

#### References

• [1] Athreya, K. B. and Karlin, S. (1968). Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Stat. 39 1801–1817.
• [2] Bergeron, F., Flajolet, P. and Salvy, B. (1992). Varieties of increasing trees. In CAAP ’92 (Rennes, 1992). Lecture Notes in Computer Science 581 24–48. Springer, Berlin.
• [3] Biggins, J. D. (1992). Uniform convergence of martingales in the branching random walk. Ann. Probab. 20 137–151.
• [4] Biggins, J. D. and Grey, D. R. (1979). Continuity of limit random variables in the branching random walk. J. Appl. Probab. 16 740–749.
• [5] Biggins, J. D. and Grey, D. R. (1997). A note on the growth of random trees. Statist. Probab. Lett. 32 339–342.
• [6] Chauvin, B., Drmota, M. and Jabbour-Hattab, J. (2001). The profile of binary search trees. Ann. Appl. Probab. 11 1042–1062.
• [7] Chauvin, B., Klein, T., Marckert, J.-F. and Rouault, A. (2003). Martingales, embedding and tilting of binary trees. Preprint.
• [8] Chauvin, B., Klein, T., Marckert, J.-F. and Rouault, A. (2005). Martingales and profile of binary search trees. Electron. J. Probab. 10 420–435.
• [9] Chauvin, B. and Rouault, A. (2004). Connecting Yule process, bisection and binary search tree via martingales. J. Iran. Stat. Soc. (JIRSS) 3 89–116.
• [10] Chen, X. (2001). Exact convergence rates for the distribution of particles in branching random walks. Ann. Appl. Probab. 11 1242–1262.
• [11] Delbaen, F., Kowalski, E. and Nikeghbali, A. (2015). Mod-$\varphi$ convergence. Int. Math. Res. Not. IMRN 11 3445–3485.
• [12] Devroye, L. (1986). A note on the height of binary search trees. J. ACM 33 489–498.
• [13] Devroye, L. (1987). Branching processes in the analysis of the heights of trees. Acta Inform. 24 277–298.
• [14] Devroye, L. and Hwang, H.-K. (2006). Width and mode of the profile for some random trees of logarithmic height. Ann. Appl. Probab. 16 886–918.
• [15] Drmota, M. (2009). Random Trees: An Interplay Between Combinatorics and Probability. Springer, Vienna.
• [16] Drmota, M. and Hwang, H.-K. (2005). Profiles of random trees: Correlation and width of random recursive trees and binary search trees. Adv. in Appl. Probab. 37 321–341.
• [17] Drmota, M., Janson, S. and Neininger, R. (2008). A functional limit theorem for the profile of search trees. Ann. Appl. Probab. 18 288–333.
• [18] Erdös, P. (1953). On a conjecture of Hammersley. J. Lond. Math. Soc. 28 232–236.
• [19] Féray, V., Méliot, P.-L. and Nikeghbali, A. (2016). Mod-$\phi$ Convergence: Normality Zones and Precise Deviations. Springer, Berlin.
• [20] Fuchs, M., Hwang, H.-K. and Neininger, R. (2006). Profiles of random trees: Limit theorems for random recursive trees and binary search trees. Algorithmica 46 367–407.
• [21] Grübel, R. and Kabluchko, Z. (2016). A functional central limit theorem for branching random walks, almost sure weak convergence and applications to random trees. Ann. Appl. Probab. 26 3659–3698.
• [22] Grübel, R. and Kabluchko, Z. (2017). Edgeworth expansions for profiles of lattice branching random walks. Ann. Inst. Henri Poincaré Probab. Stat. 53 2103–2134.
• [23] Hwang, H.-K. (2007). Profiles of random trees: Plane-oriented recursive trees. Random Structures Algorithms 30 380–413.
• [24] Jabbour-Hattab, J. (2001). Martingales and large deviations for binary search trees. Random Structures Algorithms 19 112–127.
• [25] Jacod, J., Kowalski, E. and Nikeghbali, A. (2011). Mod-Gaussian convergence: New limit theorems in probability and number theory. Forum Math. 23 835–873.
• [26] Kabluchko, Z. (2012). Distribution of levels in high-dimensional random landscapes. Ann. Appl. Probab. 22 337–362.
• [27] Kabluchko, Z., Marynych, A. and Sulzbach, H. (2016). General Edgeworth expansions with applications to profiles of random trees: Extended version. Available at http://www.math.uni-muenster.de/statistik/kabluchko/files/edgeworth_full.pdf.
• [28] Kabluchko, Z., Marynych, A. and Sulzbach, H. (2016). Mode and Edgeworth expansion for the Ewens distribution and the Stirling numbers. J. Integer Seq. 19 Art. 16.8.8.
• [29] Katona, Z. (2005). Width of a scale-free tree. J. Appl. Probab. 42 839–850.
• [30] Kowalski, E. and Nikeghbali, A. (2010). Mod-Poisson convergence in probability and number theory. Int. Math. Res. Not. IMRN 18 3549–3587.
• [31] Kowalski, E. and Nikeghbali, A. (2012). Mod-Gaussian convergence and the value distribution of $\zeta(\frac{1}{2}+it)$ and related quantities. J. Lond. Math. Soc. (2) 86 291–319.
• [32] Kuba, M. and Panholzer, A. (2007). The left-right-imbalance of binary search trees. Theoret. Comput. Sci. 370 265–278.
• [33] Mahmoud, H. M. (1992). Evolution of Random Search Trees. Wiley, New York.
• [34] Méliot, P.-L. and Nikeghbali, A. (2015). Mod-Gaussian convergence and its applications for models of statistical mechanics. In In Memoriam Marc Yor—Séminaire de Probabilités XLVII. Lecture Notes in Math. 2137 369–425. Springer, Cham.
• [35] Pittel, B. (1984). On growing random binary trees. J. Math. Anal. Appl. 103 461–480.
• [36] Régnier, M. (1989). A limiting distribution for quicksort. RAIRO Theor. Inform. Appl. 23 335–343.
• [37] Rösler, U. (1991). A limit theorem for “Quicksort”. RAIRO Theor. Inform. Appl. 25 85–100.
• [38] Schopp, E.-M. (2010). A functional limit theorem for the profile of $b$-ary trees. Ann. Appl. Probab. 20 907–950.
• [39] Sulzbach, H. (2008). A functional limit law for the profile of plane-oriented recursive trees. In Fifth Colloquium on Mathematics and Computer Science. 339–350. Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
• [40] Sulzbach, H. (2017). On martingale tail sums for the path length in random trees. Random Structures Algorithms 50 493–508.
• [41] Uchiyama, K. (1982). Spatial growth of a branching process of particles living in $\textbf{R}^{d}$. Ann. Probab. 10 896–918.