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 planeoriented random trees, as the size of the tree goes to infinity. All these models can be seen as special cases of the onesplit 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): 886918, 2006], Fuchs, Hwang and Neininger [Algorithmica, 46 (34): 367407, 2006], and Drmota and Hwang [Adv. in Appl. Probab., 37 (2): 321341, 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 arXiv:
 arXiv:1606.03920
 Bibcode:
 2016arXiv160603920K
 Keywords:

 Mathematics  Probability;
 Mathematics  Combinatorics;
 60G50 (Primary);
 60F05;
 60J80;
 60J85;
 60F10;
 60F15 (Secondary)
 EPrint:
 46 pages, 3 figures. This is an extended version of the paper to appear in the Annals of Applied Probability