On the asymptotic internal path length and the asymptotic Wiener index of random split trees

TL;DR

This paper analyzes the asymptotic behavior of the internal path length and Wiener index in random split trees, providing second order expansions and limit laws using advanced probabilistic methods.

Contribution

It introduces a second order expansion for the mean internal path length and derives limit laws for split trees using the contraction method and renewal theory.

Findings

Second order expansion for the mean internal path length

Limit law for the internal path length

Extension of methods to the Wiener index

Abstract

The random split tree introduced by Devroye (1999) is considered. We derive a second order expansion for the mean of its internal path length and furthermore obtain a limit law by the contraction method. As an assumption we need the splitter having a Lebesgue density and mass in every neighborhood of 1. We use properly stopped homogeneous Markov chains, for which limit results in total variation distance as well as renewal theory are used. Furthermore, we extend this method to obtain the corresponding results for the Wiener index.