On martingale tail sums for the path length in random trees

TL;DR

This paper establishes a central limit theorem with higher moment convergence and a law of the iterated logarithm for martingale tail sums related to the path length in various random tree models, extending previous results.

Contribution

It generalizes the CLT and LIL for martingale tail sums to a broad class of random trees, including binary search, recursive, and plane-oriented recursive trees.

Findings

Proves a CLT with higher moments convergence for the martingale tail sums.

Establishes a law of the iterated logarithm for these sums.

Applies results to multiple types of random trees.

Abstract

For a martingale $(X_n)$ converging almost surely to a random variable $X$, the sequence $(X_n - X)$ is called martingale tail sum. Recently, Neininger [Random Structures Algorithms, 46 (2015), 346-361] proved a central limit theorem for the martingale tail sum of R{\'e}gnier's martingale for the path length in random binary search trees. Gr{\"u}bel and Kabluchko [to appear in Annals of Applied Probability, (2016), arXiv 1410.0469] gave an alternative proof also conjecturing a corresponding law of the iterated logarithm. We prove the central limit theorem with convergence of higher moments and the law of the iterated logarithm for a family of trees containing binary search trees, recursive trees and plane-oriented recursive trees.