General Edgeworth expansions with applications to profiles of random trees

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(2): 886--918, 2006], Fuchs, Hwang and Neininger [Algorithmica, 46 (3--4): 367--407, 2006], and Drmota and Hwang [Adv. in Appl. Probab., 37 (2): 321--341, 2005]. 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.

[1]  Hsien-Kuei Hwang,et al.  Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees , 2006, Algorithmica.

[2]  Luc Devroye,et al.  A note on the height of binary search trees , 1986, JACM.

[3]  Philippe Flajolet,et al.  Varieties of Increasing Trees , 1992, CAAP.

[4]  K. Athreya,et al.  Embedding of Urn Schemes into Continuous Time Markov Branching Processes and Related Limit Theorems , 1968 .

[5]  J. Jacod,et al.  Mod-Gaussian convergence: new limit theorems in probability and number theory , 2008, 0807.4739.

[6]  T. Klein,et al.  Martingales and Profile of Binary Search Trees , 2004, math/0410211.

[7]  M. Drmota.  Random Trees: An Interplay between Combinatorics and Probability , 2009 .

[8]  Hsien-Kuei Hwang,et al.  WIDTH AND MODE OF THE PROFILE FOR SOME RANDOM TREES OF LOGARITHMIC HEIGHT , 2006, math/0607119.

[9]  Valentin Féray,et al.  MOD- CONVERGENCE AND PRECISE DEVIATIONS , 2014 .

[10]  J. Hammersley.  The Sums of Products of the Natural Numbers , 1951 .

[11]  A. Nikeghbali,et al.  Mod-Gaussian Convergence and Its Applications for Models of Statistical Mechanics , 2014, 1409.2849.

[12]  Zsolt Katona.  Width of a scale-free tree , 2005 .

[13]  Henning Sulzbach.  A functional limit law for the profile of plane-oriented recursive trees. , 2008 .

[14]  Eva-Maria Schopp,et al.  A functional limit theorem for the profile of b-ary trees. , 2010, 1010.3092.

[15]  Luc Devroye,et al.  Branching processes in the analysis of the heights of trees , 1987, Acta Informatica.

[16]  J. Biggins.  Uniform Convergence of Martingales in the Branching Random Walk , 1992 .

[17]  D. R. Grey,et al.  Continuity of limit random variables in the branching random walk , 1979 .

[18]  Xia Chen.  Exact Convergence Rates For The Distribution of Particles in Branching Random Walks , 2001 .

[19]  Zakhar Kabluchko,et al.  Edgeworth expansions for profiles of lattice branching random walks , 2015, 1503.04616.

[20]  P. Erdös.  On a Conjecture of Hammersley , 1953 .

[21]  Nico M. Temme,et al.  Asymptotic estimates of Stirling numbers , 1993 .

[22]  M. Drmota,et al.  The Profile of Binary Search Trees , 2001 .

[23]  Z. Kabluchko.  Distribution of levels in high-dimensional random landscapes , 2010, 1005.3207.

[24]  F. Delbaen,et al.  Mod- ϕ Convergence , 2011 .

[25]  U. Rösler.  A limit theorem for "Quicksort" , 1991, RAIRO Theor. Informatics Appl..

[26]  Jean Jabbour-Hattab,et al.  Martingales and large deviations for binary search trees , 2001, Random Struct. Algorithms.

[27]  A. Nikeghbali,et al.  Mod-Poisson convergence in probability and number theory , 2009, 0905.0318.

[28]  S. Janson,et al.  A functional limit theorem for the profile of search trees. , 2006, math/0609385.

[29]  Henning Sulzbach,et al.  Mode and Edgeworth Expansion for the Ewens Distribution and the Stirling Numbers , 2016, J. Integer Seq..

[30]  Mireille Régnier.  A Limiting Distribution for Quicksort , 1989, RAIRO Theor. Informatics Appl..

[31]  Zakhar Kabluchko,et al.  A functional central limit theorem for branching random walks, almost sure weak convergence, and applications to random trees , 2014 .

[33]  Alois Panholzer,et al.  The left-right-imbalance of binary search trees , 2007, Theor. Comput. Sci..

[34]  B. Pittel.  On growing random binary trees , 1984 .

[35]  Hsien-Kuei Hwang,et al.  Profiles of random trees: plane-oriented recursive trees (Extended Abstract) † , 2007 .

[36]  A. Rouault,et al.  Connecting Yule Process, Bisection and Binary Search Tree via Martingales , 2004, math/0410318.

[37]  Hosam M. Mahmoud,et al.  Evolution of random search trees , 1991, Wiley-Interscience series in discrete mathematics and optimization.

[38]  Henning Sulzbach,et al.  On martingale tail sums for the path length in random trees , 2014, Random Struct. Algorithms.

[39]  Hsien-Kuei Hwang,et al.  Asymptotic expansions for the Stirling numbers of the first kind , 1995 .

[40]  E. Kowalski,et al.  Mod‐Gaussian convergence and the value distribution of ζ(½ + it) and related quantities , 2009, J. Lond. Math. Soc..

[41]  V. V. Petrov.  Sums of Independent Random Variables , 1975 .

[42]  D. R. Grey,et al.  A note on the growth of random trees , 1997 .