General Edgeworth expansions with applications to profiles of random trees

TL;DR

This paper develops a general Edgeworth expansion framework for analyzing the profiles of various random trees and branching processes, leading to new insights into their mode, width, and occupation numbers.

Contribution

It introduces a unified Edgeworth expansion approach for random tree profiles and branching random walks, resolving several open problems in the field.

Findings

Derived asymptotic distributions for tree profiles

Resolved open problems on mode and width of trees

Established a general theorem for functions converging in the mod-$\

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(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.